Horofunction Compactifications and Duality

We study the global topology and geometry of the horofunction compactification of classes of symmetric spaces under Finsler distances in three settings: bounded symmetric domains of the form B=B1×⋯×Br\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B=B_1\times \cdots \times B_r$$\end{document}, where Bi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_i$$\end{document} is an open Euclidean ball in Cni\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^{n_i}$$\end{document}, with the Kobayashi distance, symmetric cones with the Hilbert distance, and Euclidean Jordan algebras with the spectral norm. For these spaces we show, that the horofunction compactification is naturally homeomorphic to the closed unit ball of the dual norm of the Finsler metric in the tangent space at the basepoint. In each case we give an explicit homeomorphism. For finite dimensional normed spaces the link between the geometry of the horofunction compactification and the dual unit ball was suggested by Kapovich and Leeb, which we confirm for Euclidean Jordan algebras with the spectral norm. Our results also show that this duality phenomenon not only occurs in normed spaces, but also in a variety of noncompact type symmetric spaces with invariant Finsler metrics.


Introduction
A well known result in the theory of manifolds of nonpositive curvature says that if M is a complete simply connected Riemannian manifold of nonpositive sectional curvature, then the horofunction compactification of M is homeomorphic to the closed unit ball of the Hilbert norm in the tangent space at the basepoint in M , e.g., [15,Proposition 1.7.6] or [16].The main goal of this paper is to establish analogues of this result for various classes of simply connected smooth manifolds with a Finsler distance.
Recall that a Finsler distance d F on a smooth manifold M has an infinitesimal form F : T M → R on the tangent bundle T M , such that d F (x, y) is the infimum of lengths, F (γ(t), γ ′ (t)) dt, over piecewise C 1 -smooth paths γ : [0, 1] → M with γ(0) = x and γ(1) = y.
More explicitly, we analyse the following general question.
Problem 1.1.Suppose M is a smooth simply connected manifold with a Finsler distance, such that the restriction of F to the tangent space T b M at b is a norm.When does there exist a homeomorphism from the horofunction compactification of M , with basepoint b, onto the dual unit ball B * 1 of the norm in T b M such that the homeomorphism maps each part in the horofunction boundary onto the relative interior of a boundary face of B * 1 ?
It should be noted that the answer to the question may depend on the basepoint b ∈ M , as the norm in T b M may have a different facial structure for different basepoints.However, the spaces we consider here are homogeneous in the sense that the facial structure of the compact convex set {v ∈ T b M : F (b, v) ≤ 1} is the same for all b ∈ M .
We confirm the existence of such a homeomorphism for a variety of manifolds in three settings: bounded convex domains in C n with the Kobayashi distance, finite dimensional normed spaces, and Hilbert geometries.For finite dimensional normed spaces the connection between the horofunction compactification and dual unit ball was suggested by Kapovich and Leeb [31,Question 6.18], who asked if for finite dimensional normed spaces the horofunction compactification is homeomorphic to the closed unit ball of the dual normed space.This was confirmed by Ji and Schilling [29,30] for polyhedral normed spaces.
For the Kobayashi distance on bounded convex domains, we consider product domains where each B i is the open unit ball of a norm with a strongly convex C 3boundary.Prime examples are polydiscs.The Finsler structure, i.e., the infinitesimal Kobayashi metric, in the tangent space at 0 is given by the norm • B whose (open) unit ball is B, see [1,Proposition 2.3.34].We show that the horofunction compactification is naturally homeomorphic to the closed ball of the dual norm of • B .For domains D ⊂ C n with the Kobayashi distance, various conditions are known that imply that the identity map on D extends as a homeomorphism from the horofunction compactification of D onto the norm closure cl D, see [5,Theorem 1.2] and [7,10,54].These conditions typically involve strong convexity and smoothness properties of the domain.In our setting, however, the domains are not smooth, and the identity does not extend as a homeomorphism, as different geodesics converging to the same point in the norm boundary of the domain can yield different horofunctions.
For finite dimensional normed spaces we will focus on the finite dimensional Euclidean Jordan algebras equipped with the spectral norm, which are precisely the finite dimensional formally real JB-algebras [4].A prime example is the real vector space Herm n (C) consisting of all n × n Hermitian matrices equipped with the norm, A = max{|λ| : λ ∈ σ(A)}.The Jordan algebra structure allows us to give a complete characterisation of the horofunctions of these normed spaces.We use this characterisation to provide a natural homeomorphism of the horofunction compactification onto the closed unit ball of the dual space.To prove the results we do not rely on the characterisation of the Busemann points in arbitrary normed spaces obtained by Walsh [49,53], but instead we exploit the Jordan algebra structure.
For Hilbert geometries (Ω, d H ) we consider domains Ω that are obtained by intersecting a symmetric cone with hyperplane.A prime example is the space of strictly-positive definite n × n Hermitian matrices with trace n.These Hilbert geometries are homogeneous in the sense that Isom(Ω) acts transitively on Ω, which ensures that the unit balls in the tangent spaces all have the same facial structure.We show, for these Hilbert geometries, that the horofunction compactification is naturally homeomorphic to the closed dual unit ball of the norm in the tangent space at the unit.We use the cone version of the Hilbert distance, see [39], which provides a convenient way to analyse its Finsler structure [44] and the dual of its norm.The horofunction compactification of these Hilbert geometries was determined in [38,Theorem 5.6] and is naturally described in terms of the Euclidean Jordan algebra associated to the symmetric cone, which will be exploited in the analysis.
The origins of the horofunction compactification go back to Gromov [6,20] who associated a boundary at infinity to any locally compact geodesic metric space.It has found numerous applications in diverse areas of mathematics including, geometric group theory [11], noncommutative geometry [46], complex analysis [1,5,7,9,10,54], Teichmüller theory [14,19,33,42,51], dynamical systems and ergodic theory [8,18,32,38] and in the study of Satake compactifications of noncompact type symmetric spaces [24,31,48].A more general set up was discussed by Rieffel [46], who recasted the horofunction compactification of a locally compact geodesic metric space as a maximal ideal space of a commutative C * -algebra.Rieffel's set up works for any metric space, but if the metric space is not proper, then the embedding into its horofunction compactification need not be a homeomorphism.
The horofunction compactification is a particularly powerful tool to study isometry groups of metric spaces and isometric embeddings between metric spaces, see [37,41,52,53].Especially useful in this context are the so called Busemann points in the horofunction compactification, which are limits of almost geodesics.They were introduced by Rieffel [46], who asked whether every horofunnction is a Busemann point in a finite dimensional normed space.Walsh [49] gave a complete solution to this problem and found necessary and sufficient conditions for a finite dimensional normed to have the property that all horofunctions are Busemann points.
In the metric spaces under consideration in this paper, all horofunctions are Busemann points.On the set of Busemann points one can define a metric known as the detour distance [2,41], which partitions the set of Busemann points into parts consisting of Busemann points that have finite detour distance to each other.So, for the metric spaces M in this paper, the horofunction compactification is the disjoint union of M and the parts in the horofunction boundary.If two Busemann points have finite detour distance, it means that the corresponding almost geodesics are in some sense asymptotic.Moreover, any isometry on the metric space M induces an isometry on the set of Busemann points under the detour distance.In each of our settings we will give an explicit homeomorphism that maps M onto the interior of the closed dual unit ball, and each part in the horofunction boundary of M onto the relative interior of a boundary face of the dual unit ball.It is this property of the homeomorphism that naturally connects the global topology of the horofunction compactification to the closed unit dual ball in each of our spaces.
In general it is hard to determine the horofunction compactification explicitly, and only in relatively few spaces has this been accomplished, even in the context of normed spaces.We give an incomplete list of results in this direction.For CAT(0) spaces the horofunction compactification is well understood, see [11,Chapter II.8] and coincides with the visual boundary.At present the horofunction compactification has been determined explicitly for a variety of normed spaces.Gutièrrez [21,22,23] computed the horofunction compactification of several classes of L p -spaces.It has also been identified for finite dimensional polyhedral normed spaces, see [12,24,30,34].In that case, the horofunction compactification is homeomorphic to closed unit ball of the dual space [29] and closely related to projective toric varieties [30].For arbitrary (possibly infinite dimensional) normed spaces the Busemann points have been characterised by Walsh [53].For Hilbert geometries there exists a characterisation of the Busemann points [51], and for the Hilbert distance on a symmetric cone in a Euclidean Jordan algebra, the horofunction compactification was obtained in [38], for the cone in a (possibly infinite dimensional) spin factor in [13], and for the p-metrics, with 1 ≤ p < ∞, on the symmetric cone in Herm n (C) in [26].

Metric geometry preliminaries
We start by recalling the construction of the horofunction compactification and the detour distance.
Let (M, d) be a metric space and let R M be the space of all real functions on M equipped with the topology of pointwise convergence.Fix a b ∈ M , which is called the basepoint, and let Lip c b (M ) denote the set of all functions The topology of pointwise convergence on Lip 1 b (M ) coincides with the topology of uniform convergence on compact sets, see [43,Section 46].In general the topology of pointwise convergence on Lip 1 b (M ) is not metrizable, and hence horofunctions are limits of nets rather than sequences.If, however, the metric space is separable, then the pointwise convergence topology on Lip 1 b (M ) is metrizable and each horofunction is the limit of a sequence.It should be noted that the embedding ι : M → Lip 1 b (M ), where ι(y) = h y , may not have a continuous inverse on ι(M ), and hence the metric compactification is not always a compactification in the strict topological sense.If, however, (M, d) is proper (i.e.closed balls are compact) and geodesic, then ι is a homeomorphism from M onto ι(M ).Recall that a map γ from a (possibly unbounded) interval The image, γ(I), is called a geodesic, and a metric space (M, d) is said to be geodesic if for each x, y ∈ M there exists a geodesic path γ : [a, b] → M connecting x and y, i.e, γ(a) = x and γ(b) = y.We call a geodesic γ([0, ∞)) a geodesic ray.
The following fact, which is slightly weaker than [46,Theorem 4.7], will be useful in the sequel.
A net (x α ) in (M, d) is called an almost geodesic net if there exists w ∈ M and for all ε > 0 there exists a β such that The notion of an almost geodesic sequence goes back to Rieffel [46] and was further developed by Walsh and co-workers in [2,37,41,53].In particular, every unbounded almost geodesic net yields a horofunction for a complete metric space [53].
exists for all z ∈ M and h ∈ ∂M h .
Given a complete metric space (M, d), a horofunction h ∈ M h is called a Busemann point if there exists an almost geodesic net (x α ) in M such that h(z) = lim α d(z, x α ) − d(b, x α ) for all z ∈ M .We denote the collection of all Busemann points by B M .Suppose that (M, d) is a complete metric space and h, h ′ ∈ ∂M h be horofunctions.Let W h be the collection of neighbourhoods of h in M h .The detour cost is given by The detour distance is given by It is known [53] that if (x α ) is an almost geodesic net converging to a horofunction h, then for all horofunctions h ′ .Moreover, on the set of Busemann points B M the detour distance is a metric where points can be at infinite distance from each other, see [53].The detour distance yields a partition of B M into equivalence classes, called parts, where h and The equivalence class of h is denoted by P h .So (P h , δ) is a metric space and B M is the disjoint union of metric spaces under the detour distance.Unlike in the setting of CAT(0) spaces, where each part is a singleton, the parts in the spaces under consideration in this paper are nontrivial.

Complex manifolds
In this section we investigate Problem 1.1 for certain bounded convex domains in C n with the Kobayashi distance.We will start by recalling some basic concepts.

Product domains and Kobayashi distance
On a convex domain D ⊆ C n the Kobayashi distance is given by for all z, w ∈ D., where ) is a proper metric space, whose topology coincides with the usual topology on C n .Moreover, (D, k D ) is a geodesic metric space containing geodesics rays, see [1,Theorem 2.6.19] or [36,Theorem 4.8.6].
For the Euclidean ball for all z, w ∈ B n , see [1, Chapters 2.2 and 2.3].
In our setting we will consider product domains B = r i−1 B i , where each B i is a open unit ball of a norm in C n i , and we will use the product property of k B , which says that where k i is the Kobayashi distance on B i , see [36,Theorem 3.1.9].So for the polydisc For the Euclidean ball, B n , it is well known that the horofunctions of (B n , k B n ), with basepoint b = 0, are given by where ξ ∈ ∂B n .Moreover, each horofunction h ξ is a Busemann point, as it is the limit induced by the geodesic ray t → e t −1 e t +1 ξ, for 0 ≤ t < ∞.Moreover, if B is a product of Euclidean balls, then the horofunctions are known, see [1, Proposition 2.4.12] and [37,Corollary 3.2].Indeed, for a product of Euclidean balls B n 1 × • • • × B nr the Kobayashi distance horofunctions with basepoint b = 0 are precisely the functions of the form, where J ⊆ {1, . . ., r} nonempty, ξ j ∈ ∂B n j for j ∈ J, and min j∈J α j = 0.Moreover, each horofunction is a Busemann point.
The form of the horofunctions of the product of Euclidean balls is essentially due to the product property of the Kobayashi distance and the smoothness and convexity properties of the balls.Indeed, more generally, the following result holds, see [37, Section 2 and Lemma 3.3].
Theorem 3.1.If D i ⊂ C n i is a bounded strongly convex domain with C 3 -boundary, then for each ξ i ∈ ∂D i there exists a unique horofunction h ξ i which is the limit of a geodesic γ from the basepoint b i ∈ D i to ξ i .Moreover, these are all horofunctions.If D = r i=1 D i , where each D i is a bounded strongly convex domain with C 3 -boundary, then each horofunction h of (D, k D ) (with respect to the basepoint b = (b 1 , . . ., b r ) is of the form, where J ⊆ {1, . . ., r} nonempty, ξ j ∈ ∂D j for j ∈ J, and min j∈J α j = 0. Furthermore, each horofunction is a Busemann point, and the part of h consists of those horofunctions h ′ with with min j∈J β j = 0. Now let D = r i=1 D i , where each D i is a bounded strongly convex domain with C 3 -boundary.Given J ⊆ {1, . . ., r} nonempty, ξ j ∈ ∂D j for j ∈ J, and α j ≥ 0 for j ∈ J with min j∈J α j = 0, we can find geodesic paths γ j : [0, ∞) → D j from b j to ξ j , and form the path γ : [0, ∞) → D, where γ(t) j = γ j (t − α j ) for all j ∈ J and t ≥ α j b j otherwise. (3.3) 3) is a geodesic path, and h γ(t) → h where h is given by (3.2).
Proof.Let k i denote the Kobayashi distance on D i .By the product property we have that for all s ≥ t ≥ 0. By construction k i (γ(s) i , γ(t) i ) ≤ k i (γ i (s), γ i (t)) = s − t for all i and s ≥ t ≥ 0. For j ∈ J with α j = 0 we have that k j (γ(s) j , γ(t) j ) = k j (γ j (s), γ j (t)) = s − t for all s ≥ t ≥ 0, and hence which shows that h γ(t) → h.
We will now define a map ϕ B : B h → B * 1 and show in the remainder of this section that it is a homeomorphism.For z ∈ B let For a horofunction h given by (3.2) we define In fact, we will prove the following theorem.
To show that ϕ B is injective on B, we need the following basic calculus fact, which can also be found in [29].For completeness we include the proof.Lemma 3.7.If µ : R r → R is given by µ(x 1 , . . ., x r ) = r i=1 e x i + e −x i , then x → ∇ log µ(x) is injective on R r .
Proof.For 0 < t < 1 we let p = 1/t ≥ 1 and q = 1/(1 − t) ≥ 1.Then by Hölder's inequality we have that for all i and some fixed C > 0. This is equivalent to ±x i = ±y i + log C for all i, and hence we have equality if and only if x = y.Thus, x → log µ(x) is a strictly convex function on R r .By strict convexity we have that log µ(x) − log µ(y) > log µ(y Combining the inequalities, we see that 0 > (∇ log µ(y) − ∇ log µ(x)) • (x − y) for all x = y, and hence x → ∇ log µ(x) is injective on R r .
Note that (∇ log µ(x)) j = e x j − e −x j r i=1 e x i + e −x i for all j.
Lemma 3.8.The map ϕ B is a continuous bijection from B onto int B * 1 .
Note that α j p(z * j ) = 0 if and only if z j = 0, and, β j p(w * j ) = 0 if and only if w j = 0. Thus, z j = 0 if and only if w j = 0. Now suppose that z j = 0, so w j = 0. Then p(v j ), ϕ B (z) = p(v j ), ϕ B (w) for each v j ∈ B j .This implies that and hence α j z * j = β j w * j .It follows that α j = β j and z * j = w * j .Thus z j = µ j w j for some µ j > 0. As α i = β i for all i ∈ {1, . . ., r}, we know by Lemma 3.7 that k j (z j , 0) = k j (w j , 0), and hence As ϕ B is injective and continuous on B, it follows from Brouwer's domain invariance theorem that ϕ B (B) is an open subset of int B * 1 by Lemma 3.6.Suppose, by way of contradiction, that 1 , which would imply that int B * 1 is the disjoint union of the nonempty open sets ϕ B (B) and its complement contradicting the connectedness of int Using the product property, k B (z n , 0) = max i k i (z n i , 0), we may assume after taking subsequences that Letting n → ∞, the righthand side converges to We now analyse ϕ B on ∂B h .
Let h ∈ ∂B h be given by h(z) = max j∈J (h ξ j (z j ) − α j ).Then To prove injectivity let h, h ′ ∈ ∂B h , where h is as in (3.2) and We have that J which is impossible.In the other case a contradiction can be derived in the same way.Now suppose there exists k ∈ J such that as cl B k is smooth and strictly convex.This is impossible.The other case goes in the same way.Thus, J = J ′ and ξ j = η j for all j ∈ J.
It follows that for all k ∈ J.We now show that α k = β k for all k ∈ J by using ideas similar to the ones used in the proof of Lemma 3.7.Let ν : R J → R be given by ν(x) = j∈J e −x j .Then for x, y ∈ R J and 0 < t < 1 we have that and we have equality if and only if there exists a constant c such that As min j∈J α j = 0 = min j∈J β j , we can conclude that α k = β k for all k ∈ J.This shows that h = h ′ and hence ϕ B is injective on ∂B h .
To complete the proof note that ϕ B (h) is in the relative open boundary face F ({ξ j ∈ ∂B j : j ∈ J}) of B * 1 .Moreover, h ′ given by (3.4) is in the same part as h if, and only if, J = J ′ and ξ j = η j for all j ∈ J by [37, Propositions 2.8 and 2.9].So, ϕ B (h ′ ) lies in F ({ξ j ∈ ∂B j : j ∈ J}) if and only if h ′ lies in the same part as h.

Continuity and the proof of Theorem 3.4
We now show that ϕ B is continuous on B h .
Proposition 3.10.The map ϕ B : where h is given by (3.2).To show that ϕ B (z n ) → ϕ B (h) we show that every subsequence of (ϕ B (z n )) has a subsequence converging to ϕ B (h).So, let (ϕ B (z n k )) be a subsequence.Then we can take a further subsequence (z n k,m ) such that (1)
by the product property of k B .As h = h ′ , we know by [37, Propositions 2.8 and 2.9] that J = J ′ , ξ j = η j and α j = β j for all j ∈ J.We also know by Lemma 2.
for z ∈ B. We show that every subsequence of (ϕ B (h n )) has a convergent subsequence with limit ϕ B (h).So let (ϕ B (h n k )) be a subsequence.Then we can take a further subsequence (ϕ B (h km )) to get that (1) There exists J 0 ⊆ {1, . . ., r} such that J km = J 0 for all m.
Note that for each j ∈ J 0 we have that h η km j → h η j in B h j , as the identity map on clB j , that is This implies that J ′ = J and η j = ξ j and β j = α j for all j ∈ J, as otherwise δ(h, h ′ ) = 0 by [37, Proposition 2.9 and Lemma 3.3].This implies that β km j → α j and η km j → ξ j for all j ∈ J ′ .Moreover, by definition β km j → ∞ for all j ∈ J 0 \ J ′ .Thus, which completes the proof.
The proof of Theorem 3.4 is now straightforward.
Proof of Theorem 3.4.It follows from Lemmas 3.8 and 3.9 and Proposition 3.10 that ϕ B : Hausdorff, we conclude that ϕ B is a homeomorphism.Moreover, ϕ B maps each part of ∂B h onto the relative interior of a boundary face of B * 1 by Lemma 3.9.

Finite dimensional normed spaces
Every finite dimensional normed space (V, • ) has a Finsler structure.Indeed, if we let where the infimum is taken over all C 1 -smooth paths γ : [0, 1] → V with γ(0) = x and γ(1) = y.So for normed spaces V the unit ball in the tangent space T b V is the same for all b ∈ V .We are interested in the following more explicit version of Problem 1.1, which was posed by Kapovich and Leeb [31,Question 6.18].Problem 4.1.For which finite dimensional normed spaces (V, • ) does there exist a homeomorphism ϕ V from the horofunction compactification of (V, • ) onto the closed dual unit ball B * 1 of V , which maps each part of the horofunction boundary onto the relative interior of a boundary face of B * 1 ?
We show that such a homeomorphism exists for Euclidean Jordan algebras equipped with the spectral norm.So we will consider the Euclidean Jordan algebras not as inner-product spaces, but as an order-unit space, which makes it a finite dimensional formally real JB-algebra, see [4,Theorem 1.11].We will give an explicit description of the horofunctions of these normed spaces and identify the parts and the detour distance.In our analysis we make frequent use of the theory of Jordan algebras and order-unit spaces.For the reader's convenience we will recall some of the basic concepts.Throughout the paper we will follow the terminology used in [3,4] and [25].

Preliminaries
Order-unit spaces A cone V + in a real vector space V is a convex subset of V with λV + ⊆ V + for all λ ≥ 0 and V + ∩ −V + = {0}.The cone V + induces a partial ordering ≤ on V by x ≤ y if y − x ∈ V + .We write x < y if x ≤ y and x = y.The cone V + is said to be Archimedean if for each x ∈ V and y ∈ V + with nx ≤ y for all n ≥ 1 we have that x ≤ 0. An element u of V + is called an order-unit if for each x ∈ V there exists λ ≥ 0 such that −λu ≤ x ≤ λu.The triple (V, V + , u), where V + is an Archimedean cone and u is an order-unit, is called an order-unit space.An order-unit space admits a norm, which is called the order-unit norm, and we have that − x u u ≤ x ≤ x u u for all x ∈ V .The cone V + is closed under the order-unit norm and u ∈ int V + .A linear functional ϕ on an order-unit space is said to be positive if ϕ(x) ≥ 0 for all x ∈ V + .It is called a state if it is positive and ϕ(u) = 1.The set of all states is denoted by S(V ) and is called the state space, which is convex set.In our case, the order-unit space is finite dimensional, and hence S(V ) is compact.The extreme points of S(V ) are called the pure states.
The dual space V * of an order-unit space V is a base norm space, see [3,Theorem 1.19].More specifically, V * is an ordered normed vector space with cone and the unit ball of the norm of V * is given by Jordan algebras Important examples of order-unit spaces come from Jordan algebras.A Jordan algebra (over R) is a real vector space V equipped with a commutative bilinear product • that satisfies the identity A basic example is the space Herm n (C) consisting of n × n Hermitian matrices with Jordan product Throughout the paper we will assume that V has a unit, denoted u.For x ∈ V we let L x be the linear map on V given by L x y = x • y.A finite dimensional Jordan algebra is said to be Euclidean if there exists an inner-product A Euclidean Jordan algebra has a cone V + = {x 2 : x ∈ V }.The interior of V + is a symmetric cone, i.e., it is self-dual and Aut(V + ) = {A ∈ GL(V ) : A(V + ) = V + } acts transitively on the interior of V + .In fact, the Euclidean Jordan algebras are in one-to-one correspondence with the symmetric cones by the Koecher-Vinberg theorem, see for example [25].
The algebraic unit u of a Euclidean Jordan algebra is an order-unit for the cone V + , so the triple (V, V + , u) is an order-unit space.We will consider the Euclidean Jordan algebras as an order-unit space equipped with the order-unit norm.These are precisely the finite dimensional formally real JB-algebras, see [4,Theorem 1.11].In the analysis, however, the inner-product structure on V will be exploited to identify V * with V .
Throughout we will fix the rank of the Euclidean Jordan algebra V to be r.In a Euclidean Jordan algebra each x can be written in a unique way as x = x + − x − , where x + and x − are orthogonal element x + and x − in V + , see [4,Proposition 1.28].This is called the orthogonal decomposition of x.
Given x in a Euclidean Jordan algebra V , the spectrum of x is given by σ(x) = {λ ∈ R : λu − x is not invertible}, and we have that V + = {x ∈ V : σ(x) ⊂ [0, ∞)}.We write Λ(x) = inf{λ : x ≤ λu} and note that Λ(x) = max{λ : λ ∈ σ(x)}, so that If, in addition, p is non-zero and cannot be written as the sum of two non-zero idempotents, then it is said to be a primitive idempotent.The set of all primitive idempotent is denoted J 1 (V ) and is known to be a compact set [28].Two idempotents p and q are said to be orthogonal if p •q = 0, which is equivalent to (p|q) = 0.According to the spectral theorem [25, Theorem III.1.2],each x has a spectral decomposition, x = r i=1 λ i p i , where each p i is a primitive idempotent, the λ i 's are the eigenvalues of x (including multiplicities), and p 1 , . . ., p r is a Jordan frame, i.e., the p i 's are mutually orthogonal and Throughout the paper we will fix the inner-product on V to be where tr(x) = r i=1 λ i and x = r i=1 λ i p i is the spectral decomposition of x.For x ∈ V we denote the quadratic representation by U x : V → V , which is the linear map, In case of a Euclidean Jordan algebra U x is self-adjoint, (U x y|z) = (y|U x z).
We identify V with V * using the inner-product.So, S(V ) = {w ∈ V + : (u|w) = 1} which is a compact convex set, as V is finite dimensional.Moreover, the extreme points of S(V ) are the primitive idempotents, see [25,Proposition IV.3.2].The dual space (V, • * u ) is a base norm space with norm, z * u = sup{(x|z) : x ∈ V with x u = 1}.If V is a Euclidean Jordan algebra, it is known that the (closed) boundary faces of the dual ball where p and q are orthogonal idempotents, see [17,Theorem 4.4].

Summary of results
To conveniently describe the horofunction compactification V h of (V, • u ), where V is a Euclidean Jordan algebra, we need some additional notation.Throughout this section we will fix the basepoint b ∈ V to be 0. Let p 1 , . . ., p r be a Jordan frame in V .Given I ⊆ {1, . . ., r} nonempty, we write p I = i∈I p i and we let V (p I ) = U p I (V ).For convenience we set p ∅ = 0, so V (p ∅ ) = {0}.
Recall [25, Theorem IV.1.1]that V (p I ) is the Peirce 1-space of the idempotent p I : which is a subalgebra.Given z ∈ V (p I ), we write Λ V (p I ) (z) to denote the maximal eigenvalue of z in the subalgebra V (p I ).
The following theorem characterises the horofunctions in V h .
Theorem 4.2.Let p 1 , . . ., p r be a Jordan frame, I, J ⊆ {1, . . ., r}, with I ∩J = ∅ and I ∪J nonempty, and α ∈ R I∪J such that min{α i : i ∈ I ∪ J} = 0.The function h : V → R given by, is a horofunction, where we use the convention that if I or J is empty, the corresponding term is omitted from the maximum.Each horofunction in V h is of the form (4.2) and a Busemann point.
To conveniently describe the parts and the detour distance we introduce the following notation.Given orthogonal idempotents p I and p J we let V (p I , p J ) = V (p I ) + V (p J ), which is a subalgebra of V with unit p IJ = p I + p J .The subspace V (p I , p J ) can be equipped with the variation norm, which is a semi-norm on V (p I , p J ).The variation norm is, however, a norm on the quotient space V (p I , p J )/Rp IJ .Theorem 4.3.Given horofunctions h and h ′ , where we have that (i) h and h ′ are in the same part if and only if p I = q I ′ and p J = q J ′ .
(ii) If h and h ′ are in the same part, then δ(h, h ′ ) = a − b var , where a = i∈I α i p i + j∈J α j p j and b = i∈I ′ β i q i + j∈J ′ β j q j in V (p I , p J ).
We will show that the following map is a homeomorphism from for x = r i=1 λ i p i ∈ V , and for h ∈ ∂V h given by (4.2).
We should note that ϕ is well defined.To verify this assume that the horofunction h given by (4.2) is represented as Then it follows from Theorem 4.3 that p I = q I ′ and p J = q J ′ .Moreover, as δ(h, h) = 0, we have that a = i∈I α i p i + j∈J α j p j = i∈I ′ β i q i + j∈J ′ β j q j = b, as min{α i : Using the map v ∈ V → e −v we deduce that i∈I e −α i p i + (u − p I ) = i∈I ′ e −β i q i + (u − q I ′ ), and hence i∈I e −α i p i = i∈I ′ e −β i q i .Likewise j∈J e −α j p j = j∈J ′ e −β j q j .We also find that and hence ϕ(h) is well defined.
We will not only prove that ϕ : V h → B * 1 is homeomorphism, but also show that ϕ maps each part of the horofunction boundary onto the relative interior of a boundary face of the dual unit ball.Indeed, we will establish the following theorem.Theorem 4.5.Given a Euclidean Jordan algebra (V, • u ), the map ϕ : V h → B * 1 is a homeomorphism.Moreover the part P h , with h given by (4.2), is mapped onto the relative interior of the closed boundary face conv (

Horofunctions
In this subsection we will prove Theorem 4.2.We first make some preliminary observations.Note that x ≤ λu if and only if 0 ≤ λu − x, which by the Hahn-Banach separation theorem is equivalent to (λu − x|w) ≥ 0 for all w ∈ S(V ).As the state space is compact, we have for each As • u is the JB-algebra norm, x • y u ≤ x u y u , see [4,Theorem 1.11].It follows that if Thus, we have the following lemma.
Lemma 4.6.If x n → x and y n → y in (V, • u ), then U x n y n → U x y.
We will also use the following technical lemma several times.
Lemma 4.7.For n ≥ 1, let p n 1 , . . ., p n r be a Jordan frame in V and I ⊆ {1, . . ., r} nonempty.Suppose that (i) p n i → p i for all i ∈ I.
Proof.We will show that every subsequence of (Λ )) be a subsequence.By (4.7) there exists By taking subsequences we may assume that Using the Peirce decomposition with respect to the Jordan frame p n k i , i ∈ I, in V (p n k I ), we can write Note that as d n k ≥ 0, we have that µ n k i = (d n k |p n k i ) ≥ 0 for all i ∈ I.We claim that for each i ∈ I \ I ′ we have that µ n k i → 0. Indeed, as I ′ is nonempty, there exist l ∈ I ′ and a constant C > 0 such that As → ∞ for all i ∈ I \ I ′ , we conclude from the previous two inequalities that µ n k i → 0 for all i ∈ I \ I ′ .Using the Peirce decomposition with respect to the Jordan frame p i , i ∈ I, we write We now show that and hence d ∈ V (p I ′ ).Note that We conclude that µ n k i → µ i for all i ∈ I, and hence (d|p j ) = µ j = 0 for all j ∈ I \ I ′ .This implies by [25, III, Exercise 3] that d • p j = 0 for all j ∈ I \ I ′ .So, which shows that d lj = 0 = d jm for all l < j < m, as they are all orthogonal.This implies (4.8).
Next we show that lim k→∞ Λ As U p I ′ d = d and U p I ′ is self-adjoint, we find that for all k large, as (U This shows that (U To complete the proof we show that As (d|p I ′ ) = (d|p I ) = 1, we know that d ∈ S(V p I ′ ), and hence we get from by (4.7) that then by definition of d n k we get for all k large that and hence (4.10) holds.
To prove that all horofunctions in V h are of the form (4.2), we first establish the following proposition by using the previous lemma.
Proposition 4.8.Let (y n ) be a sequence in V , with y n = r i=1 λ n i p n i .Suppose that h y n → h ∈ ∂V h and (y n ) satisfies the following properties: (1) There exists 1 ≤ s ≤ r such that |λ n s | = r n for all n, where r n = y n u .
(2) p n k → p k for all 1 ≤ k ≤ r.
(3) There exist I, J ⊆ {1, . . ., r} disjoint with I ∪ J nonempty, and α ∈ R I∪J with min{α i : i ∈ I ∪ J} = 0 such that r n − λ n i → α i for all i ∈ I, r n + λ n j → α j for all j ∈ J, and Then h satisfies (4.2).
Proof.Take x ∈ V fixed.Note that for all n ≥ 1, As h is a horofunction, y n u = r n → ∞ by Lemma 2.1.Thus, λ n i → ∞ for all i ∈ I and λ n j → −∞ for all j ∈ J. Now suppose that J is nonempty.Then by Lemma 4.7.We conclude that if I and J are both nonempty, then To complete the proof it remains to show that lim n→∞ x − y n u − y n u = lim n→∞ Λ(−x + y n − r n u) if J is empty, and lim n→∞ x − y n u − y n u = lim n→∞ Λ(x − y n − r n u) if I is empty.Suppose that I empty, so J is nonempty.For each i ∈ {1, . . ., r} we have that ) for all n, and hence we conclude that x − y n u − y n u = Λ(x − y n − r n u) for all n sufficiently large, and hence The argument for the case where J is empty goes in the same way.
The following corollary shows that each horofunction is of the form (4.2).
Corollary 4.9.If h is a horofunction in V h , then there exist a Jordan frame p 1 , . . ., p r in V , disjoint subsets I, J ⊆ {1, . . ., r}, with I ∪ J nonempty, and α ∈ R I∪J with min{α i : and y n u → ∞ by Lemma 2.1.
To show that the limit is equal to (4.2) it suffices to show that we can take a subsequences of (y n ) that satisfies the conditions in Proposition 4.8.First we note that by the spectral theorem [25, Theorem III.1.2],there exist for each n ≥ 1 a Jordan frame p n 1 , . . ., p n r in V and λ n 1 , . . ., λ n r ∈ R such that , where r is the rank of V .Denote r n = y n u = max i |λ n i |.Now by taking subsequences we may assume that there exist I + ⊆ {1, . . ., r} and 1 ≤ s ≤ r such that for each n ≥ 1 we have r n = |λ n s | and λ n i > 0 for all i ∈ I + and λ n i ≤ 0 for all i ∈ I + .
Now for each i ∈ {1, . . ., r} and n ≥ 1 define Again by taking subsequences we may assume that as n → ∞, for all i.Recall that α n s = 0 for all n, so α s = 0. Furthermore we may assume that p n i → p i in J 1 (V ) for all i, as it is a compact set [28].Note that p 1 , . . ., p r is a Jordan frame in V .Now let I = {i : α i < ∞ and i ∈ I + } and J = {j : α j < ∞ and j ∈ I + }.
So I ∩ J is empty, s ∈ I ∪ J and min{α i : i ∈ I ∪ J} = α s = 0. Then the subsequence of (y n ) satisfies the conditions in Proposition 4.8, and hence h is a horofunction of the form (4.2).
The next proposition shows that each function of the form (4.2) can be realised as a horofunction, and is a Busemann point.Proposition 4.10.Let p 1 , . . ., p r be a Jordan frame in V .Given I, J ⊆ {1, . . ., r}, with I ∩ J = ∅ and I ∪ J nonempty, and α ∈ R I∪J with min{α i : , where Then (y n ) is an almost geodesic sequence and h y n → h where h satisfies (4.2) for all x ∈ V .In particular, h is a Busemann point in V h .
Proof.We will use Proposition 4.8.Let k ≥ max{α i : i ∈ I ∪ J} and note that for n ≥ k we have that r n = y n u = n, as min{α i : i ∈ I ∪ J} = 0.The sequence (y n ), where n ≥ k, satisfies the conditions in Proposition 4.8.Indeed, for n ≥ k we have that r n − λ n i = α i for all i ∈ I, r n + λ n i = α i for all i ∈ J, and r n − λ n i = n otherwise.Also for 1 ≤ s ≤ r with α s = min{α i : i ∈ I ∪ J}, we have that Finally to see that (h y n ) converges, we note that if we define z = i∈I −α i p i + j∈J α j p j and w = i∈I p i − j∈J p j , then y n = nw + z, which lies on the straight-line t → tw + z.Hence (y n ) is an almost geodesic sequence, so exists of all x ∈ V .Thus, we can apply Proposition 4.8 and conclude that h satisfies (4.2).Moreover, as (y n ) is an almost geodesic sequence, h is a Busemann point in the horofunction boundary.
Combining the results so far we now prove Theorem 4.2.
Proof of Theorem 4.2.Corollary 4.9 shows that each horofunction in V h is of the form (4.2).It follows from Proposition 4.10 that any function of the form (4.2) is a horofunction and by the second part of that proposition each horofunction is a Busemann point.

Parts and the detour metric
In this subsection we will identify the parts in the horofunction boundary of V h , derive a formula for the detour distance, and establish Theorem 4.3.We begin by proving the following proposition.
Proposition 4.11.If and are horofunctions with p I = q I ′ and p J = q J ′ , then h and h ′ are in the same part and where a = i∈I α i p i + j∈J α j p j and b = i∈I ′ β i q i + j∈J ′ β j q j in V (p I , p J ).
Proof.As in Proposition 4.10, for n ≥ 1 let where where By Proposition 4.10 we know that (y n ) and (w n ) are almost geodesic sequences with h y n → h and h w n → h ′ .Note that for all m, so that Thus, In the same way it can be shown that So, it follows from (2.2) that Interchanging the roles of h and h ′ gives and hence δ(h, h ′ ) = a − b var .
To show that h and h ′ are in different parts, if p I = q I ′ or p j = q J ′ , we need the following lemma.
Lemma 4.12.If p and q are idempotents in V with p q, then U p q < p.
Proof.We have that U p q ≤ U p u = p.In fact, U p q < p. Indeed, if U p q = p, then and hence U p (u − q) = 0.This implies that p + (u − q) ≤ u by [27, Lemma 4.2.2],so that p ≤ q.This is impossible, as p q, and hence U p q < p.
Proof.Suppose that p I = q I ′ .Then p I q I ′ or q I ′ p I .Without loss of generality assume that p I q I ′ .Let (y n ) in V (p I ) and (w n ) in V (q I ′ ) be as in Proposition 4.10, so h y n → h and h w m → h ′ .To prove the statement in this case, we use (2.2) and show that Note that As w m ≤ w m u q I ′ for all m large, we have that U p I w m ≤ w m u U p I q I ′ for all m large.Thus, for all m large.We know from Lemma 4.12 that p I − U p I q I ′ > 0. As p I − U p I q I ′ ∈ V (p I ) we also have that p I − U p I q I ′ = s j=1 γ j r j , where γ j > 0 for all j and the r j 's are orthogonal idempotents in V (p I ).It now follows that for all m large, The right-hand side goes to ∞ as m → ∞, and hence (4.13) holds.
For the case p J = q J ′ a similar argument can be used.
We now prove Theorem 4.3.

The homeomorphism onto the dual unit ball
In this subsection we show Theorem 4.5.To start we prove a basic lemma that will be useful in the sequel.
We will show that ϕ given by (4.5) and (4.6) is a continuous bijection from V h onto B * 1 .As V h is compact and B * 1 is Hausdorff, we can then conclude that ϕ is a homeomorphism.We begin by showing that ϕ maps V into the interior of B * 1 .Lemma 4.15.For each x ∈ V we have that ϕ(x) ∈ int B * 1 .Proof.For x ∈ V there exists y ∈ V with y u = 1 such that where (v|w) = tr(v • w).So, if x has spectral decomposition x = r i=1 λ i p i , then we can consider the Peirce decomposition of y, Lemma 4.16.The map ϕ is injective on V .
Proof.Suppose that x, y ∈ V with x = r i=1 σ i p i and y = r i=1 τ i q i , where σ 1 ≤ . . .≤ σ r and τ 1 ≤ . . .≤ τ r , satisfy ϕ(x) = ϕ(y).Then ϕ(x) = r i=1 α i p i = r i=1 β i q i = ϕ(y).where Note that α i = α j if and only if σ i = σ j , and, Let (y n ) in V be such that ϕ(y n ) → z and write y n = r i=1 λ n i p n i .As ϕ is continuous on V , we may assume that r n = y n u → ∞.Furthermore, after taking a subsequence, we may assume that (y n ) satisfies the conditions in Proposition 4.8.So, using the notation as in Proposition 4.8, we get that The right-hand side converges to But this implies that z ∈ ∂B * 1 , which is impossible.Indeed, if we let p I = i∈I p i and p J = j∈J p j , then 1 ≥ z * u ≥ (z|p For simplicity we denote the (closed) boundary faces of B * 1 by where p and q are orthogonal idempotents in V .
Lemma 4.18.If h is a horofunction given by (4.2), then ϕ maps P h into relint F p I ,p J .
Proof.Clearly, ϕ(h) ∈ F p I ,p J if h is given by (4.2).So, ϕ maps P h into F p I ,p J by Theorem 4.3(i).To show that ϕ maps P h into relint F p I ,p J , it suffices to show that ϕ(h) ∈ relint F p I ,p J .
To do this we first consider w = (|I| + |J|) −1 (p I − p J ) ∈ F p I ,q J and show that w ∈ relintF p i ,q J .Let c ∈ F p I ,p J be arbitrary.Note that we can write c = i∈I ′ λ i q i − j∈J ′ λ j q j , where i∈I ′ q i = p I , j∈J ′ q j = p J , and i∈I ′ λ i + j∈J ′ λ j = 1 with 0 ≤ λ i , λ j ≤ 1 for all i and j.We see that w + ε(w − c) = (1 + ε)w − εc ∈ F p I ,p j for all ε > 0 small, and hence w ∈ relint F p i ,p j .
Clearly, ϕ(h) ∈ F p I ,p J = conv ((U p I (V ) ∩ S(V )) ∪ (U p J (V ) ∩ −S(V ))).To complete the proof we argue by contradiction.So suppose that ϕ(h) ∈ relintF p I ,p J .Then ϕ(h) is in the (relative) boundary of F p I ,p J , and hence for all ε > 0, as w ∈ relintF p I ,p J and F p I ,p J is convex.However, for each i ∈ I we have that the coefficient of p i in z ε , is strictly positive for all ε > 0 sufficiently small.Likewise, for each j ∈ J we have that the coefficient of −p j in z ε , is strictly positive for all ε > 0 sufficiently small.This implies that z ε ∈ F p I ,p J for all ε > 0 small, which is impossible.This completes the proof.
Using the previous results we now show that ϕ is injective on V h .
Corollary 4.19.The map ϕ : Proof.We already saw in Lemmas 4.15 and 4.16 that ϕ maps V into int B * 1 and is injective on V .So by the previous lemma, it suffices to show that if ϕ(h) = ϕ(h ′ ) for horofunctions h ∼ h ′ , then h = h ′ .Let h be given by (4.2) and suppose that h ′ is given by As min k α k = 0 = min k β k , it follows from the spectral theorem [25, Theorem III.1.2]that As each x ∈ V can be written in a unique way as x = x + − x − , where x + and x − are orthogonal element x + and x − in V + , see [4, Proposition 1.28], we find that i∈I e −α i p i = i∈I ′ e −β i q i and j∈J e −α j p i = j∈J ′ e −β j q j .This implies that and hence h = h ′ .
The next result shows that ϕ is continuous on ∂V h .
Theorem 4.20.The map ϕ : Proof.Clearly ϕ is continuous on V .Suppose (y n ) is a sequence in V with h y n → h ∈ ∂V h .We wish to show that ϕ(y n ) → ϕ(h).Let (ϕ(y n k )) be a subsequence.We will show that it has a subsequence which converges to ϕ(h).
As h is a horofunction, we know that r n = y n k u → ∞ by Lemma 2.1.For each k there exists a Jordan frame q n k 1 , . . ., q n k r in V and λ n k 1 , . . ., λ n k r ∈ R such that By taking a subsequence we may assume that there exists I + ⊆ {1, . . ., r} and 1 ≤ s ≤ r such that Note that β n k ≥ 0 for all i and k, and β n k s = 0 for all k.By taking a further subsequence we may assume that β n k i → β i ∈ [0, ∞] and q n k i → q i for all i.Let I ′ = {i ∈ I + : β i < ∞} and J ′ = {j ∈ I + : β j < ∞}.Note that s ∈ I ′ ∪ J ′ and we can apply Proposition 4.8 to conclude that h y n k → h ′ ∈ ∂V h , where As h y n k → h, we know that h = h ′ and hence δ(h, h ′ ) = 0.This implies that p I = q I ′ and p J = q J ′ by Theorem 4.3.Moreover,

It follows that
β j q j , so that i∈I e α i p i = i∈I ′ e β i q i and j∈J e α j p j = j∈J ′ e β j q j .We conclude that From Lemmas 4.15 and 4.18 we know that ϕ maps V into int B * 1 and ∂V Suppose h is given by (4.2) and for each n the horofunction h n is given by where I n , J n ⊆ {1, . . ., r} are disjoint, I n ∪ J n is nonempty, and min{β n k : k ∈ I n ∪ J n } = 0. To prove the assertion we show that each subsequence of (ϕ(h n )) has a convergent subsequence with limit ϕ(h).Let (ϕ(h n k )) be a subsequence.By taking a subsequences we may assume that (1) There exist I 0 , J 0 ⊆ {1, . . ., r} disjoint with I 0 ∪ J 0 nonempty such that I n k = I 0 and J n k = J 0 for all k. (2) (3) There exists i * ∈ I 0 ∪ J 0 such that β n k i * = 0 for all k.Let I ′ = {i ∈ I 0 : β i < ∞} and J ′ = {j ∈ J 0 : β j < ∞}, and note that i * ∈ I ′ ∪ J ′ .
Using Lemma 4.7 we now show that h n k → h ′ , where Note that if I ′ is nonempty, then by Lemma 4.7 we have that x → U q I 0 x by Lemma 4.6 and Thus, if I ′ and J ′ are both nonempty (4.15) holds.Now suppose that I ′ is empty, so J ′ is nonempty.As −x ≤ x u u, we get that On the other hand, h n k (x) ≥ − x u for all k.Thus, for all k sufficiently large, we have that which implies that (4.15) holds if I ′ is empty.In the same way it can be shown that (4.15) holds if J ′ is empty.As h n → h, we know that h ′ = h, so δ(h, h ′ ) = 0.It follows from Theorem 4.3 that p I = q I ′ , p J = q J ′ , and i∈I α i p i + j∈J α j p j = i∈I ′ β i q i + j∈J ′ β j q j .This implies that i∈I α i p i = i∈I ′ β i q i and j∈J α j p j = j∈J ′ β j q j , so that i∈I e α i p i = i∈I ′ e β i q i and j∈J e α j p j = j∈J ′ e β j q j .Thus, which completes the proof.
Theorem 4.21.The map ϕ : Proof.From Lemma 4.17 we know that ϕ(V As B * 1 is the disjoint union of the relative interiors of its faces, see [47,Theorem 18.2], we know that there exist orthogonal idempotents p I and p J such that z ∈ relintF p I ,p J .So we can write where p I = i∈I p i , q J = j∈J q j , 0 < λ k ≤ 1 for all k ∈ I ∪ J, and k∈I∪J λ k = 1. Define Then h, given by , is a horofunction by Proposition 4.10.Moreover, and hence ϕ(h) = z, which completes the proof.
The proof of Theorem 4.5 is now straightforward.
Proof of Theorem 4.5.It follows from Theorems 4.20 and 4.21 and Corollary 4.19 that ϕ : V h → B * 1 is a continuous bijection.As V h is compact and B * 1 is Hausdorff, we conclude that ϕ is a homeomorphism.It follows from Lemma 4.18 that ϕ maps parts onto the relative interior of a boundary face of B * 1 .
Remark 4.22.It is interesting to note that a similar idea can be used to show that the horofunction compactification of a finite dimensional normed space (V, • ) with a smooth, strictly convex, norm is homeomorphic to the closed dual unit ball.Indeed, in that case the horofunctions are given by h : z → −x * (z), where x * ∈ V * has norm 1, see for example [22,Lemma 5.3].Moreover for (y n ) in V we have that h y n → h if and only if y n / y n → x and y n → ∞.
In this case we define a map ψ : V h → B * 1 as follows.For x ∈ V with x = 0, let where x * ∈ V * is the unique functional with x * (x) = x and x * = 1, and let ψ(0 for all n, which contradicts h n → h.This shows that ψ is continuous bijection, and hence a homeomorphism, as V h is compact and B * 1 is Hausdorff.More generally, one can consider product spaces V = r i=1 V i with norm x V = max r i=1 v i i , where each (V i , • i ) is a finite dimensional normed space with a smooth, strictly convex, norm.In that case we have by [37,Theorem 2.10] that the horofunctions of V are given by where J ⊆ {1, . . ., r} nonempty, min j∈J α j = 0, ξ * j ∈ V * j with ξ * j = 1, and h ξ * j (v j ) = −ξ * j (v j ).One can use similar ideas as the ones in Section 3 to show that the horofunction compactification is homeomorphic to the closed unit dual ball of V .Indeed, one can define a map ϕ V : and ϕ V (0) = 0.Here p(v * i ) = (0, . . ., 0, v * i , 0, . . ., 0) and v * i is the unique functional such that v * i (v i ) = v i i and v * i i = 1, if v i = 0, and we set p(v * i ) = 0, if v i = 0.For a horofunction h given by (4.16) we define Following the same line of reasoning as in Section 3 one can prove that ϕ V is a homeomorphism.
The connection between the global topology of the horofunction compactification and the dual unit ball seems hard to establish for general finite dimensional normed spaces, and might not even hold.In the settings discussed in this paper all horofunctions are Busemann points, but there are normed spaces with horofunctions that are not Busemann, see [49].It could well be the case that the horofunction compactification of these spaces is not homeomorphic to the closed unit dual ball, but no counter example is known at present.

Hilbert geometries
In this section we study global topology and geometry of the horofunction compactification of certain Hilbert geometries.Recall that the Hilbert distance is defined as follows.Let A be a real finite dimensional affine space.Consider a bounded, open, convex set Ω ⊆ A. For x, y ∈ Ω, let ℓ xy denote the straight-line through x and y in A, and denote the points of intersection of ℓ xy and ∂Ω by x ′ and y ′ , where x is between x ′ and y, and y is between x and y ′ , as in Figure 1. for all x = y in Ω, and ρ H (x, x) = 0 for all x ∈ Ω.The metric space (Ω, ρ H ) is called the Hilbert geometry on Ω.These metric spaces generalise Klein's model of hyperbolic space and have a Finsler structure, see [44,45].In our analysis we will work with Birkhoff's version of the Hilbert metric, which is defined on a cone in an order-unit space in terms of its partial ordering.This provides a convenient way to work with the Hilbert distance and its Finsler structure.In the next subsection we will recall the basic concepts involved in our analysis.Throughout we will follow the terminology used in [39, Chapter 2], which contains a detailed discussion of Hilbert geometries and some their applications.We refer the reader to [45] for a comprehensive account of the theory of Hilbert geometries.

Preliminaries and Finsler structure
Let (V, V + , u) be a finite dimensional order-unit space.So V + is a closed cone in V with u ∈ int V + .Recall that the cone V + induces a partial ordering on V by x ≤ y if y − x ∈ V + , see Section 4.1.For x ∈ V and y ∈ V + , we say that y dominates x if there exist α, β ∈ R such that αy ≤ x ≤ βy.In that case, we write M (x/y) = inf{β ∈ R : x ≤ βy} and m(x/y) = sup{α ∈ R : αy ≤ x}.
We also note that if A ∈ GL(V ) is a linear automorphism of V + , i.e., A(V + ) = V + , then x ≤ βy if, and only if, Ax ≤ βAy.It follows that M (Ax/Ay) = M (x/y) and m(x/y) = m(Ax/Ay).
If w ∈ int V + , then w dominates each x ∈ V , and we define |x| w = M (x/w) − m(x/w).
One can verify that | • | w is a semi-norm on V , see [39,Lemma A.1.1],and a genuine norm on the quotient space V /Rw, as |x| w = 0 if and only if x = λw for some λ ∈ R.
Clearly, if x, y ∈ V are such that y = 0 and y dominates x, then x = 0, as V + is a cone.On the other hand, if y ∈ V + \ {0}, and y dominates x, then M (x/y) ≥ m(x/y).The domination relation yields an equivalence relation on V + by x ∼ y if y dominates x and x dominates y.The equivalence classes are called the parts of V + .As V + is closed, one can check that {0} and int V + are parts of V + .The parts of a finite dimensional cone are closely related to its faces.Indeed, if V + is the cone of a finite dimensional order-unit space, then it can be shown that the parts correspond to the relative interiors of the faces of V + , see [39,Lemma 1.2.2].Recall that a face of a convex set S ⊆ V is a subset F of S with the property that if x, y ∈ S and λx + (1 − λ)y ∈ F for some 0 < λ < 1, then x, y ∈ F .
It is easy to verify that if x, y ∈ V + \ {0}, then x ∼ y if, and only if, there exist 0 < α ≤ β such that αy ≤ x ≤ βy.Furthermore, if x ∼ y, then m(x/y) = sup{α > 0 : y ≤ α −1 x} = M (y/x) −1 . (5.2) Birkhoff 's version of the Hilbert distance on V + is defined as follows: for all x ∼ y with y = 0, d H (0, 0) = 0, and d H (x, y) = ∞ otherwise.Note that d H (λx, µy) = d H (x, y) for all x, y ∈ V + and λ, µ > 0, so d H is not a distance on V + .It is, however, a distance between pairs of rays in each part of V + .In particular, if ϕ : V → R is a linear functional such that ϕ(x) > 0 for all x ∈ V + \ {0}, then d H is a distance on It is worth noting that any Hilbert geometry can be realised as (Ω V , d H ) for some order-unit space V and strictly positive linear functional ϕ.
A Hilbert geometry (Ω V , d H ) has a Finsler structure, see [44].Indeed, if one defines the length of a piecewise C 1 -smooth path γ : [0, 1] → Ω V by where the infimum is taken over all piecewise C 1 -smooth paths in Ω V with γ(0) = x and γ(1) = y.So for Hilbert geometries Problem 1.1 can be formulated more explicitly as follows.
Problem 5.2.Let (V, V + , u) be a finite dimensional order-unit space and ϕ : V → R be a linear functional with ϕ(x) > 0 for all x ∈ V + \ {0} and ϕ(u) = 1.For which Hilbert geometries (Ω V , d H ) does there exists a homeomorphism from the horofunction compactification Ω It should be noted that in the case of Hilbert geometries the unit ball {x ∈ V /Rw : |x| w ≤ 1} in the tangent space at w ∈ Ω V may have a different facial structure for different w.This phenomenon appears frequently in the case where Ω V is a polytope.This problem, however, does not arise in the spaces we analyse here.Indeed, we will consider order-unit spaces (V, V + , u), where V is a Euclidean Jordan algebra of rank r, V + is the cone of squares, and u is the algebraic unit.So int V + is a symmetric cone and Isom(Ω V ) acts transitively on Ω V .Throughout we will take ϕ : V → R with ϕ(x) = 1 r tr(x), which is a state and In that case we call (Ω V , d H ) a symmetric Hilbert geometry.A prime example is In a symmetric Hilbert geometry the distance can be expressed in terms of the spectrum.Indeed, we know that for x ∈ V invertible, the quadratic representation U It follows that Moreover, for each w ∈ Ω V we have that which shows that the facial structure of the unit ball in each tangent space is identical, as U w −1/2 is an invertible linear map.By using the Jordan algebra structure there is a direct way to show that a symmetric Hilbert geometry has a Finsler structure.Proposition 5.3.If (Ω V , d H ) is a symmetric Hilbert geometry, then for each x, y ∈ Ω V we have that d H (x, y) = inf L(γ), where the infimum is taken over all piecewise C 1 -smooth paths γ : [0, 1] → Ω V with γ(0) = x and γ(1) = y, and for all n ≥ m ≥ 1.Thus, (v n ) is an almost geodesic sequence in Ω V , and hence each horofunction in Ω h V is a Busemann point.
To identify the parts and describe the detour distance we need the following general lemma.
On the other hand, if w does not dominate v, then Assume, by way of contradiction, that (λ n ) is bounded.Then λ n → λ * < ∞, since (λ n ) is increasing, and λ n w n −v → λ * w−v ∈ V + , as V + is closed.This contradicts (5.7), and hence Before we identify the parts in ∂Ω h V and the detour distance, it is useful to recall the following fact: M (x/y) = M (y −1 /x for x ∈ Ω V .The following assertions hold: (i) h and h ′ are in the same part if and only if y ∼ y ′ and z ∼ z ′ .
(ii) If h and h ′ are in the same part, then δ(h, h Proof.Consider the spectral decompositions: y = i∈I λ i p i , z = j∈J µ j p j , y ′ = i∈I ′ α i q i , and z ′ = j∈J ′ β j q j .Set Then h yn → h and h wn → h ′ by the proof of [38,Theorem 5.6].For all n ≥ 1 large we have that w n u = y ′ u = 1, so that Clearly w n+1 ≤ w n and w n → y ′ .Also So, for all n ≥ 1 large, we have that w −1 Interchanging the roles between h and h ′ we find that H(h, h ′ ) = ∞ if y does not dominate y ′ , or, z does not dominate z ′ , and The characterisation of the parts and the detour distance is a more explicit description of the general one one given in [41,Theorem 4.9] in the case of symmetric Hilbert geometries.

The homeomorphism
Let us now define a map ϕ H : Ω , .
We note that ϕ H (h) is well-defined by Proposition 5.6.We will prove the following theorem in the sequel.We first analyse the dual unit ball B * 1 of | • | u and its facial structure.The following fact, see also [40, Section 2.2], will be useful.
Lemma 5.8.Given an order-unit space (V, V + , u), the norm |•| u on V /Ru coincides with the quotient norm of 2 • u on V /Ru.
Proof.Denote the quotient norm of 2 • u on V /Ru by • q .Then Recall that in a Euclidean Jordan algebra V each x has a unique orthogonal decomposition x = x + − x − , where x + and x − are orthogonal elements in V + , see [4,Proposition 1.28].Let So the dual unit ball B * 1 in Ru ⊥ is given by see [3,Theorem 1.19], and its (closed) boundary faces are precisely the nonempty sets of the form, where p and q are orthogonal idempotents, see [17,Theorem 4.4].
To prove Theorem 5.7 we collect a number of preliminary results.
Lemma 5.9.For each x ∈ Ω V we have that ϕ H (x) ∈ int B * 1 , and for each h ∈ ∂Ω h V we have that ϕ H (h) ∈ ∂B * 1 .
To prove the second assertion let h be a horofunction given by h(x) = log M (y/x)+log M (z/x −1 ), where y u = z u = 1 and (y|z) = 0. Write y = i∈I α i q i and z = j∈J β j q j .If we now let q I = i∈I q i and q J = j∈J q j , then −u ≤ q I − q J ≤ u and Moreover, for each −u ≤ w ≤ u we have that Combining the inequalities shows that ϕ H (h) ∈ ∂B * 1 .
To prove injectivity of ϕ H on Ω V we need the following lemma, which is similar to Lemma 3.7.
Lemma 5.10.Let µ i : R r → R, for i = 1, 2, be given by Proof.For 0 < t < 1, p = 1/t and q = 1/(1 − t) we have, by Hölder's inequality, that = µ 1 (x) t µ 1 (y) 1−t , and we have equality if and only if there exists a C 1 > 0 such that e y i = (e (1−t)y i ) q = C 1 (e tx i ) p = C 1 e x i for all i, which is equivalent to y i = x i + c 1 for all i.Likewise, µ 2 (tx + (1 − t)y) = µ 2 (x) t µ 2 (y) 1−t and we have equality if and only if y i = x i + c 2 for all i.
Proof.Suppose that ϕ H (x) = ϕ H (y), where x = r i=1 λ i p i and y = r i=1 µ i q i in Ω V .Note that 0 < λ i , µ i for all i and (x|u) = tr(x) = r = tr(y) = (y|u).After possibly relabelling we can write ϕ H (x) = Proof.We know from Lemma 5.9 that ϕ H maps ∂Ω h V into ∂B * 1 .To prove that it is onto let w ∈ ∂B * 1 .Then there exists a face, say A p,q = 2conv ((U p (V ) ∩ S(V )) ∪ (U q (V ) ∩ −S(V ))) ∩ Ru ⊥ where p and q are orthogonal idempotents, such that w is in the relative interior of A p,q , as B * 1 is the disjoint union of the relative interiors of its faces [47,Theorem 18.2].So, w = i∈I α i p i − j∈J β j q j , where α i > 0 for all i ∈ I, β j > 0 for all j ∈ J, and i∈I α i + j∈J β j = 2.Moreover, i∈I p i = p and j∈J q j = q.
As w ∈ Ru ⊥ , we have that 0 = (u|w) = i∈I α i − j∈J β j , and hence i∈I α i = j∈J β j = 1.Put α * = max i∈I α i and β * = max j∈J β j .Furthermore, for i ∈ I set λ i = α i /α * and for j ∈ J set µ j = β j /β * .Then w = i∈I α i p i k∈I α k − j∈J β j q j k∈J β k = i∈I λ i p i k∈I λ k − j∈J µ j q j k∈J µ k .Note that 0 < λ i ≤ 1 for all i ∈ I and max i∈I λ i = 1.Likewise, 0 < µ j ≤ 1 for all j ∈ J and max j∈J β j = 1.Now let y = i∈I λ i p i and z = j∈J µ j q j .Then y u = z u = 1 and (y|z) = 0. Furthermore, if we let h : Ω V → R be given by h(x) = log M (y/x) + log M (z/x −1 ) for x ∈ Ω V , then h is a horofunction by Theorem 5.4 and This completes the proof.
We already saw in Lemma 5.11 that ϕ H is injective on Ω V .The next lemma shows that ϕ H is injective on Ω h V .
for x ∈ Ω V .By taking a further subsequence we may assume that v m → v ∈ ∂V + and w m → w ∈ ∂V + .Then v u = w u = 1 and (v|w) = 0.Moreover, log M (v m /x) → log M (v/x) and log M (w m /x −1 ) → log M (w/x −1 ), for each x ∈ Ω V , as y → M (y/x) is a continuous map on V , see [38,Lemma 2.2].Thus, h km → h * ∈ ∂Ω This completes the proof of the continuity of ϕ H . Thus, ϕ H is a continuous bijection from Ω As Ω h V is compact and B * 1 is Hausdorff, we conclude that ϕ H is a homeomorphism.
To complete the proof of the theorem it remains to show that ϕ H maps each part onto the relative interior of a boundary face of B * 1 .Let h(x) = log M (y/x) + log M (z/x −1 ) be a horofunction, where y = i∈I λ i p i and z = j∈J µ j p j with λ i , µ j > 0 for all i ∈ I and j ∈ J. Let p I = i∈I p i and p J = j∈J p j .As ϕ H is surjective, it suffices to show that ϕ H maps P h into the relative interior of So, let h ′ ∈ P h where h ′ (x) = log M (y ′ /x) + log M (z ′ /x −1 ) for x ∈ Ω V .Then p I ∼ y ∼ y ′ and p J ∼ z ∼ z ′ .Using the spectral decomposition write y ′ = i∈I ′ α i q i and z ′ = j∈J ′ β j q j , where α i > 0 for all i ∈ I ′ and β j > 0 for all j ∈ J ′ .Now let q I ′ = i∈I ′ q i and q J ′ = j∈J ′ q j .It follows that p I ∼ q I ′ and p J ∼ q J ′ .So, face(p I ) = face(q I ′ ) and face(p J ) = face(q J ′ ).As face(p I ) ∩ [0, u] = [0, p I ] and face(q I ′ ) ∩ [0, u] = [0, q I ′ ] by [4,Lemma 1.39], we conclude that p I = q I ′ .In the same way we get that p J = q J ′ .As α i > 0 for all i ∈ I ′ and β j > 0 for all j ∈ J ′ , we have that ϕ H (h ′ ) = y ′ tr(y ′ ) − z ′ tr(z ′ ) is in the relative interior of A q I ′ ,q J ′ = A p I ,p J .

Final remarks
It would be interesting to find a general class of simply connected smooth manifolds M with a Finsler distance for which Problem 1.1 has a positive solution.A common feature of the spaces considered in this paper is the property that the facial structure of the unit ball {x ∈ T b M : F (b, v) ≤ 1} is the same for all b ∈ M .In particular, one could consider spaces where the d F -isometry group of M acts transitively on M .This is the case for all normed spaces and the symmetric Hilbert geometries.A second feature of the spaces considered here is that all horofunctions arise as limits of geodesics.This property might be a useful further assumption to make.
Even if both these properties hold in a finite dimensional normed space or a Hilbert geometry, then it is not clear how one can define a homeomorphism for these spaces, despite the fact that we know all horofunctions by Walsh [49,52].What made things work in our settings was the Jordan algebra structure and its associated spectral theory, which allowed us to give a more explicit description of the horofunctions and the parts of the horofunction boundary that gave a clear link with the dual norm.
It is also worth noting that if both M and the normed space (T b M, • b ) at the basepoint b have a positive solution to Problem 1.1, then there exists a homeomorphism between the horofunction compactifications of these spaces that maps parts onto parts.It would be interesting to know if this connection exists more generally.More specifically, one can ask the following general question.Problem 6.1.Suppose M is a simply connected smooth manifold with a Finsler distance, such that the restriction of F to the tangent space T b M at b is a norm.When does there exist a homeomorphism between the horofunction compactification of M with basepoint b and the horofunction compactification of the normed space (T b M, • b ), which maps parts onto parts?
A solution to this problem would allow one to study the horofunction compactifications of these manifolds by analysing the horofunction compactifications of finite dimensional normed spaces, which might be easier.

hV
with basepoint u onto the closed dual unit ball B * 1 of | • | u on V /Ru, which maps each part of the horofunction boundary onto the relative interior of a boundary face of B * 1 ?

Theorem 5 . 7 .
If (Ω V , d H ) is a symmetric Hilbert geometry, then the map ϕ H : Ω h V → B *1 is a homeomorphism which maps each part of ∂Ω h V onto the relative interior of a boundary face of B * 1 .

→ i∈I λ i p i k∈I λ k − j∈J µ j p j k∈J µ j = w .
Now let w * = i∈I p i − j∈J p j and note that −u ≤ w * ≤ u, as (y|z) = 0. w ∈ ∂B * 1 , which is a contradiction.Lemma 5.13.The map ϕ H maps ∂Ω h V onto ∂B * 1 .
′ ) = ϕ H (h ′ ).Using the fact that the orthogonal decomposition of an element in V is unique, see [ h * (x) = log M (v/x) + log M (w/x −1 ), by Theorem 5.4.As h n → h, we have that h = h * .This implies that y = v and z = w, as otherwise δ(h, h * ) = 0 by Proposition 5.6.Thus, v m → y and w m → z, and henceϕ H (h km ) = v m tr(v m ) − w m tr(w m ) → y tr(y) − z tr(z)= ϕ H (h).
where each B i is an open unit ball of a norm in C n i .Then B is the open unit ball of the norm • B on C n .In fact, C n : Re w, z ≤ 1 for all w ∈ clB}.Now suppose that each B i is strictly convex and smooth.Then the closed ball B * 1 has extreme points p(ξ * i ) = (0, . . ., 0, ξ * i , 0, . . ., 0), where ξ * i ∈ C n i is the unique supporting functional at ξ i ∈ ∂B i , i.e., Re ξ i , ξ * i = 1 and Re w i , ξ * i < 1 for w i ∈ cl B i with w i = ξ i .The relatively open faces of B * 1 are the sets of the form: Theorem 3.4.If B = r i=1 B i , where each B i is the open unit ball of a norm on C n i which is strongly convex and has a C 3 -boundary, then ϕ B : B The map ϕ B : injectivity and surjectivityThroughout the remainder of this section we assume that B = r i=1 B i and each B i is the open unit ball of a norm on C n i , which is strongly convex and has a C 3 -boundary.So for each ξ i ∈ ∂B i there exists a unique ξ * i ∈ C n i such that Re ξ i , ξ * i = 1 and Re w i , ξ * i < 1 for all w i ∈ cl B i with w i = ξ i , 1−t .Moreover, equality holds if and only if and only if τ i = τ j , as ∇ log µ(x) is injective.It now follows from the spectral theorem (version 1) [25, Theorem III.1.1]that x = y.Lemma 4.17.The map ϕ maps V onto int B * 1 .Proof.As ϕ is continuous on V and ϕ(V ) ⊆ int B * 1 it follows from Brouwer's domain invariance theorem that ϕ(V ) is open in int B * 1 .Suppose, for the sake of contradiction, that ϕ(V ) = int B * 1 .Then ∂ϕ(V ) ∩ int B * 1 is nonempty, as otherwise ϕ(V ) is closed and open, which would imply that int B * 1 is the disjoint union of two nonempty open sets contradicting the connectedness of int B * 1 .So we can find a and ψ is continuous on int B * 1 .To show continuity on ∂V h , we assume, by way of contradiction, that (h n ) is a sequence of horofunctions with h n → h and h n (z) = −x * n (z) for all z ∈ V , and there exists a neighbourhood U of ψ(h) in B * 1 such that ψ(h n ) ∈ U for all n.Then for each z * ∈ ∂B * 1 with z * ∈ U we have that z * (x) < 1.So, by compactness Theorem 5.1.(Ω V , d H ) is a metric space and d H