1 Introduction

Given a bounded open convex subset \(\Omega \) of a Euclidean space, David Hilbert proposed in [13] a natural metric \(H_\Omega (x, y)= H(x,y)\) on \(\Omega \), now called the Hilbert metric. It is defined for \(x\not = y\) in \(\Omega \) as the logarithm of the cross ratio of the quadruple \((x, y, b(x, y), b(y,x))\), where \(b(x, y)\) is the point where the ray from \(x\) through \(y\) hits the boundary of \(\Omega \). This metric is Finslerian and projective. We recall that a Finslerian metric on \(\Omega \) (or, more generally, on a differentiable manifold) is determined by a generalized norm on each tangent space in such a way that the distance between two points in \(\Omega \) is equal to the infimum of lengths of paths joining them. Here, the length of a path is computed by integrating along it “norms” of vectors tangent to the path, and the term “norm” refers to a nonnegative sub-additive positively homogeneous function on each tangent space called the Finsler norm or the Minkowski functional of the metric. We shall say that a metric on a subset of Euclidean space is projective if Euclidean straight lines are images of geodesics for that metric.

The open unit disc in \(\mathbb R ^2\) equipped with its Hilbert metric is a prominent example of a projective metric, since it is Klein’s model of hyperbolic \(2\)-space. As a matter of fact, this metric was the motivating example for Hilbert when he defined (what we call now) the Hilbert metric on a general convex set.

The value \(H(x, y)\) of the Hilbert metric on a bounded open convex subset \(\Omega \) of \(\mathbb R ^n\) can be written, for \(x\not = y\), as

$$\begin{aligned} \log \frac{|x-b(x,y)||y-b(y,x)|}{|y-b(x,y)||x-b(y,x)|} = \log \frac{|x-b(x,y)|}{|y-b(x,y)|} + \log \frac{|y-b(y,x)|}{|x-b(y,x)|}. \end{aligned}$$

Funk [11] looked at the first term of the right hand side of the above equation as a kind of metric, even though it is not symmetric in \(x\) and \(y\). This is now called the Funk metric. The reader is referred to Funk’s paper [11] and to the papers [18, 19] for some background on the Funk metric.

More generally, given a set \(X\), we can consider functions \(\delta : X \times X \rightarrow \mathbb R _+ \cup \{ \infty \} \) satisfying the following two properties:

  1. (1)

    \(\delta (x,x) = 0\) for all \(x\) in \(X\) ;

  2. (2)

    \(\delta (x, z) \le \delta (x, y) + \delta (y, z)\) for all \(x,y\) and \(z\) in \(X\).

In [18], such a function is called a weak metric. Note that in this definition neither the symmetry (\(\delta (x, y) = \delta (y,x)\)) nor the nondegeneracy \(\left( \delta (x, y) = 0 \Rightarrow x=y\right) \) is assumed. A weak metric may be Finslerian in the sense that it may be induced by a Minkowski functional on each tangent space, in the same way as a genuine metric may be Finslerian.

For brevity, we shall call a weak metric a metric, and when such a function does or does not satisfy the other axioms satisfied by a metric, we shall mention this explicitly whenever this is needed.

We recall that Hilbert’s Fourth Problem, from the collection of mathematical problems he presented in 1900 at the Second International Congress of Mathematicians in Paris, is entitled: “Problem of the straight line as the shortest distance between two points.” Hilbert elaborates on this statement in [14], and he also places it in a historical perspective. Mathematicians now agree on the fact that Hilbert’s formulation of the problem is rather vague, which makes this problem (like several other problems in Hilbert’s list) admit various different precise formulations and therefore several possible solutions. The most common formulation of that problem (which is close to the original one) is in two parts, namely, (1) to characterize the metrics on subsets of Euclidean space for which the Euclidean straight lines are geodesics, and (2) to study such metrics individually. We note by the way that the axioms of a metric space as we intend them today were not yet formulated at the time Hilbert proposed his problems (they were given later on by Maurice Fréchet in his thesis, in 1907). It was clear from the beginning that Hilbert’s problem might also concern generalized metrics which do not satisfy the symmetry axiom, like the Funk metric which we study in this paper.

Herbert Busemann, who spent much of his career engaged with Hilbert’s problem IV, formulated it as follows (see [8]): “The fourth problem concerns the geometries in which the ordinary lines, i.e. lines of an n-dimensional (real) projective space \(\mathbb P ^n\) or pieces of them are the shortest curves or geodesics. Specifically, Hilbert asks for the construction of these metrics and the study of the individual geometries.”

For general convex subsets of hyperbolic space and of the sphere, we shall see that the geodesics of the underlying space are geodesics for the Funk and the Hilbert metrics of the convex sets. With these examples in mind, it is natural to address the analogue of Hilbert’s Problem IV in the non-Euclidean settings, that is, the problem of characterizing and of studying individually the metrics on subsets of the sphere and on subsets of hyperbolic space for which the spherical and the hyperbolic straight lines respectively are geodesics. In the case of the Hilbert metric, these problems amount in some sense to the Euclidean problem. This is due to the fact that despite the lack of linear structure in the hyperbolic and spherical geometries, when it comes to the geometry of Hilbert metrics, one can still capture the projective geometry in a manner almost identical to the one in the Euclidean situation. This holds however only in the case of Hilbert metrics, and not in that of Funk metrics. We mention in this respect that in a recent preprint [2, Álvarez-Paiva shows that there are no interesting geodesically reversible Finsler metrics (that is, Finsler metrics where every oriented geodesic equipped with the reverse orientation can be reparametrized so that it is also geodesic) defined on a compact rank one symmetric space (e.g. the sphere). This contrasts with the case of the Funk metric studied in the present paper. In fact, Álvarez-Paiva stated the problem of extending the Hilbert problem to rank one symmetric spaces. Studying the Funk/Hilbert metrics on subsets of such spaces is a problem that includes what we study in the present paper.

The authors would like to thank Norbert A’Campo for sharing his enthusiasm and ideas and the referees of this paper for several corrections and helpful suggestions.

2 The Funk metric in Euclidean space

In this section, we collect some known facts about the Funk and Hilbert metrics defined on convex subsets of Euclidean spaces. We take [18] as our reference for the Funk and Hilbert metrics, and we also refer to the first part of the paper [22].

Let \(\Omega \) be an open convex subset of Euclidean space \(\mathbb R ^n\). Note that we allow \(\Omega \) to be unbounded.

There are three different descriptions of the Funk metric on \(\Omega \). The first one is the original definition which we already referred to:

$$\begin{aligned} F_1(x, y) = \log \frac{d(x, b(x,y))}{d(y, b(x,y))}, \end{aligned}$$
(1)

for \(x\not = y\) in \(\Omega \), the point \(b(x,y)\) being the intersection of the Euclidean ray \(\{x+t \xi _{xy}: t>0\}\) from \(x\) though \(y\) with the boundary \(\partial \Omega \) when such an intersection point exists, \(\xi _{xy}\) being the unit tangent vector in \(\mathbb R ^n\) pointing from \(x\) to \(y\). In the case where the Euclidean ray \(\{x+t \xi _{xy}: t>0\}\) is contained in \(\Omega \) (and therefore does not intersect the boundary), we set the distance \(F_1(x, y)\) to be 0. This makes the function \((x,y)\mapsto F_1(x,y)\) defined on \(\Omega \times \Omega \) and continuous with respect to the topology on this set induced from the toplogy of \(\mathbb R ^n\).

The second description is a variational interpretation of Formula (1) using the geometry of supporting hyperplanes; we set:

$$\begin{aligned} F_2 (x,y) = \sup _{\pi \in \mathcal{P }} \log \frac{d(x, \pi )}{d(y, \pi )}, \end{aligned}$$

where \(\mathcal P \) is the set of all supporting hyperplanes of \(\Omega \). This is given in [22]. In the literature, a variational characterization of the Hilbert metric appears in the work of Birkhoff [3] and Nussbaum [16], where the supporting hyperplanes are treated instead as elements of the dual space; note that the Funk metric is not mentioned in these works.

Finally, the Finsler structure \(p_{\Omega , x}(\xi )\) of the Funk metric is given by the following function (the Minkowski functional) on vectors \(\xi \) at each tangent space to \(\Omega \) at \(x\):

$$\begin{aligned} p_{\Omega , x}(\xi ) = \sup _{\pi \in \mathcal{P }} \frac{ \langle \eta _{\pi }(x), \xi \rangle }{d(x, \pi )} \end{aligned}$$

where \(\eta _\pi \) is the unit vector at \(x\) perpendicular to the hyperplane \(\pi \) and directed towards \(\pi \).

This weak norm on each tangent space is defined so that the Funk distance is described as the infimum of lengths of curves:

$$\begin{aligned} F_3 (x,y) = \inf _{\sigma } \int \limits _a^b p_{\Omega , \sigma (t)} (\dot{\sigma }(t)) dt, \end{aligned}$$

the infimum being taken over the piecewise \(C^1\)-curves \(\sigma \) in \(\Omega \) with \(\sigma (a) = x\) and \(\sigma (b) = y\).

We emphasize that for a convex domain \(\Omega \subset \mathbb R ^n\), the three quantities \(F_1(x,y), F_2(x,y), F_3(x,y)\) are all equal to each other (see [22] for a proof) and we set

$$\begin{aligned} F(x, y) := F_1(x,y) = F_2(x,y)= F_3(x,y) \end{aligned}$$

for every \(x\) and \(y\) in \(\Omega \).

The aim of this paper is to consider Funk-like metrics in non-Euclidean geometries: hyperbolic and spherical. Formally, the exposition is very similar in the two cases. We shall first give an exposition of the theory in the case of the \(n\)-dimensional hyperbolic space and we shall then mention the changes needed for the case of spherical geometry.

3 The hyperbolic and spherical Funk metrics

3.1 The hyperbolic Funk metric

Given an open convex subset \(\Omega \) of \(\mathbb{H }^n\), we shall define a Funk-type metric (which we shall call the Funk metric of \(\Omega \)) and provide three descriptions of it corresponding to \(F_1, F_2\) and \(F_3\) of the Euclidean Funk metric case.

We shall also study the geodesics of this metric. We recall that if \((X,d)\) is a metric space, then a path \(s:I \rightarrow (X, d)\), where \(I\) is an interval of \(\mathbb R \), is said to be geodesic if for any \(a,b,c\) in \(I\) satisfying \(a\le b\le c\) we have \(d(s(a),s(b)) + d(s(b), s(c)) = d(s(a), s(c))\). In the case where the interval \(I\) is compact, then the path \(s\) (and also its image in \(X\)) is said to be a geodesic segment. In the case where \(I\) is a half-infinite interval \(I=[a,\infty [\), then \(s\) is called a geodesic ray.

We can represent the convex set \(\Omega \) as \( \cap _{\pi \in \mathcal{P }} H_{\pi }\) where \(H_{\pi }\) is the (open) half-space bounded by a hyperplane \(\pi \) touching \(\Omega \) at a boundary point and containing this convex set. In analogy with the Euclidean situation, we call the hyperplane \(\pi \) a supporting hyperplane of \(\Omega \). The index set \(\mathcal P \) is the set of all supporting hyperplanes of \(\Omega \). Given a point \(b\) of the boundary of \(\Omega \), we shall denote by \(\pi (b)\) a hyperplane touching \(\Omega \) at \(b\). That for each \(p\) in \(\partial \Omega \) there exists a supporting hyperplane \(\pi (b)\) follows from the definition of convexity of \(\Omega \). In general, there can be more than one supporting hyperplane of \(\Omega \) at \(b \in \partial \Omega \). We shall denote by \(\mathcal P (b)\) the set of all supporting hyperplanes of \(\Omega \) containing the point \(b\). The existence and the basic properties of supporting hyperplanes in the non-Euclidan setting are analogous to those in the Euclidean setting. There are several good introductory texts to convexity theory in Euclidean spaces, and we refer the reader to the classical texts [10] and [9]. For later use, we denote by \(\mathcal{P }(b)\) the set of supporting hyperplanes at \(b \in \partial \Omega \). We denote by \(d\) the hyperbolic metric in \(\mathbb H ^n\).

Given two distinct points \(x\) and \(y\) in a convex set \(\Omega \), we denote by \(R(x,y)=\{ \exp _x (t \xi _{xy}) \,\, | \,\, t>0\}\) the geodesic ray starting at \(x\) and passing through \(y\) where, as in the Euclidean case, \(\xi _{xy}\) is the unit tangent vector at \(x\) of the arc-length parameterized geodesic in \(\mathbb H ^n\) connecting \(x\) and \(y\).

Definition 3.1

For a pair of points \(x\) and \(y\) in \(\Omega \subset \mathbb{H }^n\), the Funk (asymmetric) distance from \(x\) to \(y\) is defined by

$$\begin{aligned} F(x, y)= \left\{ \begin{array}{ll} \log \frac{\sinh d(x, b(x,y))}{\sinh d(y, b(x,y))} &{}\quad \text{ if }\quad x\not =y \quad \text{ and }\quad R(x,y)\cap \partial \Omega \not =\emptyset ,\\ 0 &{}\quad \text{ otherwise }\\ \end{array}\right. \end{aligned}$$

where as usual the point \(b(x,y)\) is the intersection with the boundary \(\partial \Omega \) of the hyperbolic geodesic ray \(R(x,y)\) from \(x\) though \(y\).

We shall see below that the function \(F\) satisfies the triangle inequality.

We will consider only the case where the ray \(R(x,y)\) is not contained in \(\Omega \). The other case can be dealt with easily.

We first recall a classical trigonometric identity in hyperbolic geometry (We refer to Fig. 1). For a given hyperbolic triangle \(\triangle (A,B,C)\) in the hyperbolic plane \(\mathbb{H }^2\) with respective angles \(\alpha , \beta ,\gamma \) with \(\gamma = \pi / 2\) and with side lengths \(a,b\) and \(c\) opposite to the vertices \(A, B\) and \(C\) respectively, we have

$$\begin{aligned} \sinh b = \sinh c \sin \beta . \end{aligned}$$

The formula is a special case of the hyperbolic sine rule which is recalled in the appendix.

Fig. 1
figure 1

Similarity property for right triangles

Full size image

Note that a Euclidean right triangle with corresponding labeling would satisfy \(b = c \sin \beta \) and we have here an instance of a correspondence which often occurs between the Euclidean and the hyperbolic trigonometric formulae, where the hyperbolic formulae are obtained by replacing the side lengths by the hyperbolic sines of these lengths, and similar transformations. (The sine rule is another example.) Choosing a point \(A^{\prime }\) on the side \(c\) and letting \(C^{\prime }\) be its nearest point projection on the side \(a\), we have another right triangle \(\triangle (A^{\prime },B,C^{\prime })\) with angles \(\alpha ^{\prime }, \beta ,\gamma ^{\prime } \), with \(\gamma ^{\prime }= \pi /2\), and with side lengths \(a^{\prime },b,c^{\prime }\) opposite to the vertices \(A^{\prime }, B\) and \(C^{\prime }\) respectively, satisfying

$$\begin{aligned} \sinh b^{\prime } = \sinh c^{\prime } \sin \beta . \end{aligned}$$

As the ratios \(\sinh b / \sinh c\) and \( \sinh b^{\prime } / \sinh c^{\prime } \) are equal to \(\sin \beta \), we shall say that the two triangles \(\triangle (A,B,C)\) and \(\triangle (A^{\prime },B,C^{\prime })\) are similar with the side lengths being weighted by the \(\sinh \) function.

For any \(x\not = y\) in \(\Omega \), let \(\pi _0\) be a supporting hyperplane for \(\Omega \) at \(b(x,y)\). Let \(\Pi _{\pi _0} : \mathbb H ^n \rightarrow \pi _0\) be the nearest point projection map.

First assume that \(b(x,y)\not =\Pi _{\pi _0}(x)\). Note the similarity (in the above sense) between the two right triangles \(\triangle (x, \Pi _{\pi _0}(x), b(x,y))\) and \(\triangle (y, \Pi _{\pi _0}(y), b(x,y))\). These two triangles lie in a hyperbolic plane \(\mathbb{H }^2\) isometrically embedded in \(\mathbb{H }^n\) which is uniquely determined by the three points \(x, \Pi _{\pi _0}(x), b(x,y)\). Thus, we have

$$\begin{aligned} \log \frac{\sinh d(x, b(x,y))}{\sinh d(y, b(x,y))} = \log \frac{\sinh d(x, \pi _0)}{\sinh d(y, \pi _0)}, \end{aligned}$$
(2)

by the above similarity property of triangles.

In the case where \(b(x,y)=\Pi _{\pi _0}(x)\), Eq. (2) is trivially satisfied.

Now let \(x\not = y\) be two points in \(\Omega \). Using the convexity of \(\Omega \), the quantity \(F(x, y)\) in Definition 3.1 can be characterized variationally as follows. For \(\pi \) in \(\mathcal P \) and for any nonzero tangent vector \(\xi \) at \(x\), let \(T(x,\xi , \pi )\) be the intersection point of \(\pi \) with the ray starting at \(x\) and pointing in the direction \(\xi \). Consider the case where \(\xi = \xi _{xy}\), the unit vector at \(x\) pointing from \(x\) to \(y\). When the hyperplane \(\pi \) supports \(\Omega \) at \(b(x, y)\), we have \(T(x,\xi _{xy}, \pi ) = b(x, y)\) and otherwise (when \(\pi \notin \mathcal{P }(b(x, y))\)) the point \(T(x, \xi _{xy}, \pi )\) lies outside \(\Omega \). When \(\pi \notin \mathcal{P }(b(x,y))\), by the similarity property between the triangles \(\triangle (x, \Pi _{\pi }(x), T(x, \xi _{xy}, \pi ))\) and \(\triangle (y, \Pi _{\pi }(y), T(\xi _{xy}, \pi ))\) again, we have

$$\begin{aligned} \frac{\sinh d(x, \pi )}{\sinh d(y, \pi )} =\frac{\sinh d(x, T(x,\xi _{xy},\pi ))}{\sinh d(y, T(x,\xi _{xy}, \pi ))}. \end{aligned}$$
(3)

Note that among all the choices of \(\pi \in \mathcal{P } \) which intersect the ray \(\{ x+ t \xi _{xy} | t>0 \}, T(x, \xi _{xy}, \pi )\) is closest to \(x\) and \(y\) when the hyperplane \(\pi \) supports \(\Omega \) at \(b(x, y)\). It follows that the right hand side of (3) is maximized when \(\pi \) is an element of \(\mathcal{P }(b(x, y))\). This in turn says that a hyperplane \(\pi \) which supports \(\Omega \) at \(b(x,y)\) maximizes the ratio \(d(x, T(x,\xi _{xy}, \pi ))/d(y, T(x,\xi _{xy}, \pi ))\) among all the elements in \(\mathcal{P }\); that is,

$$\begin{aligned} \log \frac{\sinh d(x, b(x,y))}{\sinh d(y, b(x, y))} = \sup _{\pi \in \mathcal{P }} \log \frac{\sinh d(x, \pi )}{\sinh d(y, \pi )}. \end{aligned}$$

Hence we have the following characterization of the Funk metric \(F\) for \(\Omega \subset \mathbb{H }^n\):

Theorem 3.2

The Funk metric on a convex subset \(\Omega \subset \mathbb{H }^n\) has the following variational formulation:

$$\begin{aligned} F(x, y) = \sup _{\pi \in \mathcal{P }} \log \frac{\sinh d(x, \pi )}{\sinh d(y, \pi )}. \end{aligned}$$

With this formulation, one can readily see that \(F(x, y)\) satisfies the triangle inequality, for

$$\begin{aligned} F(x, y) + F(y, z)&= \sup _{\pi \in \mathcal{P }} \log \frac{\sinh d(x, \pi )}{\sinh d(y, \pi )} + \sup _{\pi \in \mathcal{P }} \log \frac{\sinh d(y, \pi )}{\sinh d(z, \pi )} \\&\ge \sup _{\pi \in \mathcal{P }} \left( \log \frac{\sinh d(x, \pi )}{\sinh d(y, \pi )} + \log \frac{\sinh d(y, \pi )}{\sinh d(z, \pi )} \right) \\&= \sup _{\pi \in \mathcal{P }} \log \frac{\sinh d(x, \pi )}{\sinh d(z, \pi )} \\&= F(x, z) \end{aligned}$$

Note that the triangle inequality becomes an equality when

$$\begin{aligned}&\sup _{\pi \in \mathcal{P }} \log \frac{\sinh d(x, \pi )}{\sinh d(y, \pi )} + \sup _{\pi \in \mathcal{P }} \log \frac{\sinh d(y, \pi )}{\sinh d(z, \pi )}\nonumber \\&\quad = \sup _{\pi \in \mathcal{P }} \left( \log \frac{\sinh d(x, \pi )}{\sinh d(y, \pi )} + \log \frac{\sinh d(y, \pi )}{\sinh d(z, \pi )} \right) \end{aligned}$$

is satisfied. For this to occur, we only need to have \(\mathcal{P }(b(x, y)) \cap \mathcal{P }(b(y,z)) \ne \emptyset \). Indeed, let \(\pi _0\) be an element of the set \(\mathcal{P }(b(x, y)) \cap \mathcal{P }(b(y,z)) \ne \emptyset \). Then the boundary points \(b(x,y)\) and \(b(y,z)\) share the same supporting hyperplane \(\pi _0\), and therefore

$$\begin{aligned} \sup _{\pi \in \mathcal{P }} \log \frac{\sinh d(x, \pi )}{\sinh d(y, \pi )}&= \log \frac{\sinh d(x, \pi _0)}{\sinh d(y, \pi _0)},\\ \sup _{\pi \in \mathcal{P }} \log \frac{\sinh d(y, \pi )}{\sinh d(z, \pi )}&= \log \frac{\sinh d(y, \pi _0)}{\sinh d(z, \pi _0)} \end{aligned}$$

and

$$\begin{aligned} \sup _{\pi \in \mathcal{P }} \log \frac{\sinh d(x, \pi )}{\sinh d(z, \pi )} = \log \frac{\sinh d(x, \pi _0)}{\sinh d(z, \pi _0)} \end{aligned}$$

inducing the equality. The observation is summarized in the following proposition. For \(x\) and \(y\) in \(\Omega \), we denote, as before, by \(R(x,y)\) the geodesic ray starting at \(x\) and passing through \(y\).

Proposition 3.3

Let \(\Omega \) be an open convex subset of \(\mathbb{H }^n\) such that \(\partial \Omega \) contains the image of some hyperbolic geodesic segment \(\sigma :[p, q] \rightarrow \mathbb{H }^2\) which we denote by \(\overline{pq}\) and let \(x\) and \(z\) be two distinct points in \(\Omega \) such that \(R(x,z) \cap \overline{pq} \ne \emptyset \). Let \(\Omega ^{\prime }\) be the intersection of \(\Omega \) with the hyperbolic plane \(\mathbb{H }^2\) in \(\mathbb{H }^n\) containing \(\{x\} \cup \overline{pq}\). Then, for any point \(y\) in \(\Omega ^{\prime }\) satisfying \(R(x,y) \cap \overline{pq} \ne \emptyset \) and \(R(y,z) \cap \overline{pq} \ne \emptyset \), we have \(F(x, y) + F(y,z) = F(x,z)\).

A notable situation when one has \(\mathcal{P }(b(x, y)) \cap \mathcal{P }(b(y,z)) \ne \emptyset \) is when \(x,y\) and \(z\) are collinear, meaning that they lie on a common geodesic, with \(y\) lying between \(x\) and \(z\). This in turn says that the hyperbolic geodesics are Funk geodesics, or – as Hilbert would say – that the Funk metric is projective. This result is a hyperbolic analogue of Corollary 8.2 of [18].

On the other hand, when \(\pi _0\) is in the set \(\mathcal{P }(b(x, y)) \cap \mathcal{P }(b(y,z))\) and the three points \(x,y,z\) do not lie on a geodesic, the concatenation of the geodesic segments \(\overline{xy}\) and \(\overline{yz}\) is also a Funk geodesic, a situation occurring when the boundary set \(\partial \Omega \) contains a hyperbolic geodesic segment. This statement is a hyperbolic analogue of Corollary 8.4 of [18].

We next consider the complementary situation where \(\mathcal{P }(b_1) \cap \mathcal{P }(b_2) = \emptyset \) for any pair of distinct points \(b_i \in \partial \Omega \). Geometrically this characterizes strict convexity of the domain \(\Omega \), namely the boundary \(\partial \Omega \) contains no closed line segment. From the preceding argument, it follows that the only way the equality for the triangle inequality occurs is when the three points \(x,y\) and \(z\) are collinear. Hence, for strictly convex domains, the Funk geodesics consist of line segments only. Equivalently, given a pair of distinct points in \(\Omega \), there is a unique Funk geodesic connecting them. This corresponds to Corollary 8.8 of [18].

We summarize this in the following

Proposition 3.4

Let \(\Omega \) be an open convex subset of \(\mathbb H ^n\) and let \(F(x,y)\) be its Funk metric. Then,

  1. (1)

    the hyperbolic geodesics of \(\Omega \) are also Funk geodesics;

  2. (2)

    the Funk geodesics of \(\Omega \) are hyperbolic geodesics if and only if \(\Omega \) is strictly convex, that is, if its boundary does not contain any nonempty open hyperbolic segment.

There is another proof of Proposition 3.4 that gives at the same time the triangle inequality for the Funk metric. It imitates the classical proofs of the triangle inequality given in the case of the Euclidean Funk geometry that are given in [7], p. 158 and [23], p. 85; cf. the appendix of this article.

Let us also note the following

Proposition 3.5

Any hyperbolic geodesic segment starting at a point \(x\) in \(\Omega \) and ending at a point on \(\partial \Omega \) is (the closure of the image of) a geodesic ray for the Funk metric on \(\Omega \).

Proof

This follows from the formula defining the Funk metric (Definition 3.1) and the fact that the hyperbolic geodesics in \(\Omega \) are Funk geodesics. \(\square \)

Example 3.6

Let us consider the particular example of a Funk metric where the open convex set \(\Omega \) is an ideal triangle of \(\mathbb{H }^2\). We choose the case of the ideal triangle because such a triangle exists only in hyperbolic geometry. We can model the triangle to be the region in the upper half plane bounded by the \(y\)-axis, which we call \(\pi _1\), the line \(\{x=1\}\), which we call \(\pi _2\) and the semi-circle \(\pi _3\) connecting the origin and \((1, 0)\) and which is perpendicular to the real line. The geodesic lines \(\pi _1,\pi _2,\pi _3\) are also the supporting hyperplanes of \(\Omega \). Without loss of generality, we suppose that two distinct points \(x_1\) and \(x_2\) in \(\Omega \) are located so that the hyperbolic geodesic ray from \(x_1\) through \(x_2\) hits the \(y\)-axis \(\pi _1\) at \(b(x_1, x_2)\), and hence the hyperbolic Funk distance

$$\begin{aligned} F(x_1, x_2) = \max _{\pi _1, \pi _2, \pi _3} \log \frac{\sinh d(x_1, \pi _i)}{\sinh d(x_2, \pi _i)} \end{aligned}$$

is realized by \(\log \displaystyle \frac{\sinh d(x_1, \pi _1)}{\sinh d(x_2, \pi _1)}\). Then an elementary calculation gives an explicit value of the Funk distance as

$$\begin{aligned} F(x_1, x_2) = \log \left( \frac{1- m_1^2}{1- m_2^2 } \cdot \frac{m_2}{m_1} \right) \end{aligned}$$

where \(m_i\) is the absolute value of the slope of the (Euclidean) line segment connecting \(x_i\) and the (hyperbolic) foot of \(x_i\) on the \(y\)-axis \(\pi _1\).

We next consider the infinitesimal linear structure of the Funk metric \(F\), by identifying it with a Finsler norm on the tangent spaces. We first recall the Euclidean setting. In this setting, the Funk metric is induced by a Finsler structure, the tautological weak Finsler structure in the sense of [18], given by the following Minkowski functional:

$$\begin{aligned} p_{\Omega , x}(\xi ) = \sup _{\pi \in \mathcal{P }} \frac{\Vert \xi \Vert }{d(x, T(x, \xi , \pi ))}. \end{aligned}$$
(4)

In this formula, the supremum is achieved when the supporting hyperplane \(\pi \) supports \(\Omega \) at the point where the ray \(\{x+ t \xi \, | \, t>0\}\) meets the boundary set \(\partial \Omega \). Using similarity of Euclidean triangles, this can be written as

$$\begin{aligned} p_{\Omega , x}(\xi ) = \sup _{\pi \in \mathcal{P }} \frac{\langle \xi , \eta _\pi \rangle }{d(x, \pi )} \end{aligned}$$
(5)

where \(\eta _\pi \) is the unit tangent vector at \(x\) with direction opposite to the gradient vector of the functional \(d(., \pi )\). We note that the gradient vector field’s integral curves are the geodesics which meet the supporting hyperplane perpendicularly.

The infinitesimal linear structure of the Funk metric is obtained by linearizing the following expressions for a fixed \(\pi \in \mathcal{P }\),

$$\begin{aligned} \log \frac{d(x, T(x, \xi , \pi ))}{d(\alpha (t), T(x, \xi , \pi ))} \text{ or } \text{ equivalently } \log \frac{d(x, \pi )}{d(\alpha (t), \pi )}, \end{aligned}$$

where \(\alpha :[0,\infty ) \rightarrow \mathbb{R }^n\) is a geodesic ray with \(\alpha (0)=x, \alpha (1) = y\) and \(\Vert \alpha ^{\prime }(t)\Vert = \text{ const. }\) and where \(T(x, \xi , \pi )\) is the point where the geodesic ray and the hyperplane \(\pi \) intersect. We then have

$$\begin{aligned} \frac{\Vert \alpha ^{\prime }(0) \Vert }{d(x, T(x, \alpha ^{\prime }(0), \pi ))} \text{ or } \text{ equivalently } \frac{\langle \alpha ^{\prime }(0), \eta _\pi \rangle }{d(x, \pi )}. \end{aligned}$$

In the hyperbolic space we identify the value of the Minkowski functional for the Funk metric \(F\) as

$$\begin{aligned} p_{\Omega , x}(\xi ) = \sup _{\pi \in \mathcal{P }} \frac{\cosh d(x, T(x, \xi , \pi ) ) }{\sinh d(x, T(x, \xi , \pi ))} \Vert \xi \Vert , \end{aligned}$$
(6)

or equivalently as

$$\begin{aligned} p_{\Omega , x}(\xi ) = \sup _{\pi \in \mathcal{P }} \frac{\cosh d(x, \pi ) }{\sinh d(x, \pi )} \langle \eta _{\pi }(x), \xi \rangle \end{aligned}$$
(7)

where the norm and the inner product are those of the hyperbolic metric and where \(\eta _\pi \) is the unit tangent vector at \(x\) whose direction is opposite to the one of the gradient vector of the functional \(d(., \pi )\) at \(x\). In this formula, it is easy to see that the supremum is achieved when the point \(T(x, \xi , \pi )\) coincides with a boundary point \(b \in \partial \Omega \), namely \(\pi \in \mathcal P (T(x, \xi , \pi ))\).

The two representations come from the linearizations of

$$\begin{aligned} \log \frac{\sinh d(x, T(x, \xi , \pi ))}{\sinh d(\alpha (t), T(x, \xi , \pi ))} \text{ and } \log \frac{\sinh d(x, \pi )}{\sinh d(\alpha (t), \pi )} \end{aligned}$$

respectively, where \(\alpha :[0,\infty ) \rightarrow \mathbb{H }^n\) is a geodesic ray with \(\alpha (0)=x, \alpha (1) = y\) and \(\Vert \alpha ^{\prime }(t)\Vert = \text{ const. }\) and where \(T(x, \xi , \pi )\) is the point where the geodesic ray and the hyperplane \(\pi \) intersect.

In either representation, it is easy to see that the functional is convex in \(\xi \in T_x \mathbb R ^n\), since the functional \( p_{\Omega , x}\) is convex (in fact, it is linear) in \(\xi \) for each fixed \(\pi \in \mathcal{P }\), and since by taking the supremum over \(\pi \), convexity is preserved.

Alternatively, one can see the convexity of the indicatrix \(C(x, \Omega )\), that is, the set of vectors with norm equal to one:

$$\begin{aligned} C(x, \Omega ) := \{ \xi \in T_x \mathbb{H }^n \,\, | \,\, p_{\Omega , x}(\xi ) = 1 \}, \end{aligned}$$

by noting that for \(\xi _1\) and \(\xi _2\) in \(C(x, \Omega )\), we have

$$\begin{aligned} p_{\Omega , x}\left( \frac{\xi _1 + \xi _2}{2}\right)&= \sup _{\pi \in \mathcal{P }} \frac{\cosh d(x, \pi ) }{\sinh d(x, \pi )} \left\langle \nu _{\pi }(x), \frac{\xi _1 + \xi _2}{2} \right\rangle \\&\le \frac{1}{2} \sup _{\pi \in \mathcal{P }} \frac{\cosh d(x, \pi ) }{\sinh d(x, \pi )} \langle \nu _{\pi }(x), \xi _1 \rangle + \frac{1}{2} \sup _{\pi \in \mathcal{P }} \frac{\cosh d(x, \pi ) }{\sinh d(x, \pi )} \langle \nu _{\pi }(x), \xi _2 \rangle \\&= \frac{1}{2} + \frac{1}{2} = 1. \end{aligned}$$

This says that \((\xi _1 + \xi _2)/2\) lies inside the indicatrix \(C(x, \Omega )\), hence the unit ball of the norm \(p_{\Omega , x}\) is a convex set in \(T_x \Omega \).

We now show that the infimum, among all piecewise \(C^1\)-paths \(\gamma \) with given endpoints, of the length computed with the Finsler norm \(p_{\Omega , x}\) coincides with the Funk metric. Namely:

Theorem 3.7

The Finsler distance \(d(x, y)\) induced by the Minkowski functional \(p_{\Omega , x}\) (6) or (7) coincides with the hyperbolic Funk metric \(F\) on \(\Omega \subset \mathbb{H }^n\).

Proof

For a given pair of points \(x\) and \(y\) in \(\Omega \), let \(\alpha (t) \) be the hyperbolic geodesic ray \(\exp _x t \xi _{xy}\) from \(x\) through \(y\), and let \(b(x, y) = T(x, \xi _{xy}, \pi _{b(x,y)})\) be the point where this geodesic ray hits the boundary \(\partial \Omega \) (with the notation we have been using in the Euclidean case). Then the \(F\)-length of the curve \( \alpha \) from \(x\) to \(y\) is

$$\begin{aligned} L(\alpha ) = \int \limits _0^1 p_{\Omega , \alpha (t)} (\alpha ^{\prime }(t)) dt = \log \frac{\sinh d(x, b(x, y))}{\sinh d (y, b(x, y))} = F(x, y). \end{aligned}$$

Thus, the Finsler distance \(d(x, y) = \inf _\gamma L(\gamma )\) is bounded above by the Funk- type distance \(F(x, y)\).

On the other hand, consider the convex set in \(\mathbb{H }^n\) bounded by the supporting hypersurface \(\pi _{b(x, y)}\) alone. This set has its own Funk metric,

$$\begin{aligned} F_{\pi _{b(x, y)}}(s, t) := \log \frac{\sinh d(s, \pi _{b(x, y)})}{\sinh d (t, \pi _{b(x, y)})}. \end{aligned}$$

There is a simple comparison \(F(s, t) \ge F_{\pi _{b(x, y)}}(s, t)\), as \(F(s, t)\) is the supremum of \(\displaystyle \log \frac{\sinh d(s, \pi )}{\sinh d (t, \pi )}\) over all the supporting hypersurfaces including \(\pi _{b(x, y)}\). Hence for the pair \((x, y)\), the value

$$\begin{aligned} F_{\pi _{b(x, y)}}(x, y) = \log \frac{\sinh d(x, \pi _{b(x, y)})}{\sinh d (y, \pi _{b(x, y)})} \end{aligned}$$

provides a lower bound for \(F(x, y)\).

From the equality

$$\begin{aligned} \log \frac{\sinh d(x, b(x, y))}{\sinh d (y, b(x, y))}= \log \frac{\sinh d(x, \pi _{b(x, y)})}{\sinh d (y, \pi _{b(x, y)})}, \end{aligned}$$

we conclude that \(d(x, y) = F(x, y)\). \(\square \)

We end this section by the following result, which contrasts with the case of Euclidean geometry:

Proposition 3.8

There exists a convex set \(\Omega \subset \mathbb{H }^n\) such that the Funk distance function \(F(x, y)\) is not convex in \(y\).

Proof

We start with an arbitrary convex set \(\Omega \). Consider a point \(x\) in \(\Omega \) and a constant-speed geodesic segment \(\alpha : [0,1] \rightarrow \mathbb{H }^n\) with \(\alpha (0)=y\) where \(y\) is any point in \(\Omega \). For a fixed supporting hyperplane \(\pi \) of \(\Omega \), we denote by \(\eta _\pi \) the vector field that is opposite to the gradient vector field of the functional \(d(., \pi )\). Then the first and second derivative of the Funk distance function \(F(x, \alpha (t))\) are given as:

$$\begin{aligned} \frac{d}{dt} \log \frac{\sinh d(x, \pi )}{\sinh d(\alpha (t), \pi )} \ = \frac{\cosh d(\alpha (t), \pi )}{\sinh d(\alpha (t), \pi )} \langle \dot{\alpha }(t), \eta _\pi (\alpha (t)) \rangle \end{aligned}$$

and

$$\begin{aligned} \frac{d^2}{dt^2} \log \frac{\sinh d(x, \pi )}{\sinh d(\alpha (t), \pi )} \Big |_{t=0}&= \frac{1}{\sinh ^2 d(y, \pi )} \langle \dot{\alpha }(0), \eta _\pi (x) \rangle ^2 \\&\quad + \frac{\cosh d(y, \pi )}{\sinh d(y, \pi )} \left( \frac{d}{dt} \langle \dot{\alpha }(t), \eta _\pi (\alpha (t)) \rangle \Big |_{t=0} \right) . \end{aligned}$$

Note that the quantity \(\langle \dot{\alpha }(t), \eta _\pi (\alpha (t)) \rangle \) is equal to \(\Vert \dot{\alpha }(t) \Vert \cos \theta (t)\) where \(\theta (t)\) is the angle between the velocity vector \(\dot{\alpha }(t)\) and the unit vector \(\eta _\pi (\alpha (t))\), and that the speed \(\Vert \dot{\alpha }(t) \Vert \) is constant over time. Now the negative sectional curvature of \(\mathbb{H }^n\) implies that unless \(\alpha ^{\prime }(0) = \pm \eta _\pi (\alpha (0))\), the angle \(\theta (t)\) is increasing in \(t\), which in turns implies

$$\begin{aligned} \frac{d}{dt} \langle \dot{\alpha }(t), \eta _\pi (\alpha (t)) \rangle < 0. \end{aligned}$$

We use this to show that the function \(F(x,y)\) is not always convex in \(y\). Consider the simplest convex set \(\Omega \) of the half-space, namely, the case where the boundary \(\partial \Omega \) consist of one single hyperplane \(\pi \). Choose two points \(x\) and \(y\) in \(\Omega \) so that the ray from \(x\) through \(y\) hits \(\pi \) perpendicularly. Choose \(\alpha (t)\) to be a geodesic from \(y=\alpha (0)\) with \(\alpha ^{\prime }(0)\) perpendicular to \(\eta _\pi \). Then the first term of the second derivative is zero, and the second term is strictly negative, making the function \(F(x, \alpha (t))\) in \(t\) concave at \(t=0\). \(\square \)

It is known that, in contrast, the Euclidean Funk metric \(F(x, y)\) is convex in \(y\) (see [22]).

3.2 The Funk metric in spherical geometry

We shall adapt the term hyperplane to the case of \(S^n\); this means a complete totally geodesic codimension-one subspace, that is, a great sphere. Likewise, a (closed or open) connected component in \(S^n\) bounded by a hyperplane is called a half-space.

The definition of a convex set on the sphere is slightly more delicate than the one in Euclidean or in hyperbolic space, because given any distinct two points on the sphere, there are two distinct geodesics (arcs of great circle) joining them. Even if we insist on geodesics of shortest length, for some pairs of points of the sphere (namely, for points which are diametrically opposite), there are two distinct geodesics of shortest length. For this reason, we give the following

Definition 3.9

A subset \(\Omega \) of the sphere is said to be convex if \(\Omega \) is contained in an open half-space and if every pair of points in \(\Omega \) can be connected by a geodesic which is contained in \(\Omega \).

As in the Euclidean and the hyperbolic case, given a convex set \(\Omega \) in \(S^n\), we can again represent it as \(\bigcap _{\pi \in \mathcal{P }} H_{\pi }\) where \(\mathcal{P }\) is the set of supporting hyperplanes of \(\Omega \). Again, for \(b\in \partial \Omega \), we shall denote by \(\mathcal{P }(b) \subset \mathcal{P }\) the set of supporting hyperplanes at \(b\).

Remark 3.10

In all what follows, we shall consider the sphere of radius one, that is, the space of constant curvature \(+1\). Otherwise, if the curvature is different from one, a constant factor has to be inserted in the trigonometric formulae. Then for a point \(x\) in the convex set \(\Omega \) and a hyperplane \(\pi (b)\in \mathcal{P }(b)\), note that \(d(x, \pi (b))\) is at most \(\pi /2\). We also note that unless the point \(x\) is the center of the hemisphere bounded by \(\pi (b)\), the nearest point projection of \(x\) to \(\pi (b)\) is single valued.

Now we define the Funk metric in spherical geometry. We use the angular metric on the sphere, and we shall assume that the diameter of the open convex set \(\Omega \) is \(<\pi /2\) for reasons that will become apparent immediately after the next definition.

Definition 3.11

For a pair of points \(x\) and \(y\) in \(\Omega \subset S^n\), we define the Funk (asymmetric) metric by

$$\begin{aligned} F(x, y)= \left\{ \begin{array}{ll} \log \frac{\sin d(x, b(x, y))}{\sin d(y, b(x,y))}&{}\quad \text{ if }\, x\not =y,\\ 0 &{}\quad \text{ if }\,x=y\\ \end{array}\right. \end{aligned}$$

where (as in the Euclidean and the hyperbolic cases) the point \(b(x, y)\) is the first intersection point of the boundary \(\partial \Omega \) with the geodesic ray \(\{ \exp _x t \xi _{xy} \,\, | \,\, t>0 \}\) from \(x\) through \(y\), and \(\xi _{xy}\) is the unit tangent vector at \(x\) of the arc-length parameterized geodesic connecting \(x\) and \(y\).

Note that the sine function is strictly increasing for angles between \(0\) and \(\pi /2\), and this makes the Funk distance \(F(x,y)\) always nonnegative.

We recall that for a given spherical right triangle \(\triangle (A,B,C)\) with angles \(\alpha , \beta \) and \(\gamma = \pi / 2\) and with sides (or side lengths) \(a,b\) and \(c\) opposite to the vertices \(A, B\) and \(C\) respectively, we have the formula

$$\begin{aligned} \sin b = \sin c \sin \beta . \end{aligned}$$

In analogy with the hyperbolic case, we note that choosing a point \(A^{\prime }\) on the side \(c\) and letting \(C^{\prime }\) be its nearest point projection on the side \(a\), we have another right triangle \(\triangle (A^{\prime },B,C^{\prime })\) with angles \(\alpha ^{\prime }, \beta ,\gamma ^{\prime } \), with \(\gamma ^{\prime }= \pi /2\), and with side lengths \(a^{\prime },b,c^{\prime }\) opposite to the vertices \(A^{\prime }, B\) and \(C^{\prime }\) respectively, satisfying

$$\begin{aligned} \sin b^{\prime } = \sin c^{\prime } \sin \beta . \end{aligned}$$

Again, as the ratios \(\sin b / \sin c\) and \( \sin b^{\prime } / \sin c^{\prime } \) are equal to \(\sin \beta \), we regard the two triangles \(\triangle (A,B,C)\) and \(\triangle (A^{\prime },B,C^{\prime })\) as similar with side lengths being weighted by the function \(\sin \).

Following the same argument as in hyperbolic geometry, we have the following variational formula for the Funk metric:

Theorem 3.12

The Funk metric on a convex subset \(\Omega \subset S^n\) is also given by:

$$\begin{aligned} F(x, y) = \sup _{\pi \in \mathcal{P }} \log \frac{\sin d(x, \pi )}{\sin d(y, \pi )}. \end{aligned}$$

For the sphere, we identify the value of the Minkowski functional for the Funk metric \(F\) as

$$\begin{aligned} p_{\Omega , x}(\xi ) = \sup _{\pi \in \mathcal{P }} \frac{\cos d(x, T(x, \xi , \pi ) ) }{\sin d(x, T(x, \xi , \pi ))} \Vert \xi \Vert , \end{aligned}$$
(8)

or, equivalently, as

$$\begin{aligned} p_{\Omega , x}(\xi ) = \sup _{\pi \in \mathcal{P }} \frac{\cos d(x, \pi ) }{\sin d(x, \pi )} \langle \eta _{\pi }(x), \xi \rangle \end{aligned}$$
(9)

where \(\eta _\pi \) is the unit tangent vector at \(x\) whose direction is opposite to the one of the gradient vector of the functional \(d(., \pi )\) at \(x\). These expressions appear naturally by following the same argument we have seen for the hyperbolic Funk metric. We then have the following statement for the spherical Funk metric, the proof of which is also almost identical to the analogous one in the hyperbolic case.

Theorem 3.13

The Finsler distance \(d(x, y)\) induced by the Minkowski functional \(p_{\Omega , x}\) (8) or (9) coincides with the spherical Funk metric \(F\) on \(\Omega \subset S^n\).

3.3 Convexity of \(F(x, y)\) in the \(y\)-variable

Consider a point \(x\) in \(\Omega \) and a geodesic segment \(\alpha : [0,1] \rightarrow S^n\) with \(\alpha (0)=y\) where \(y\) is any point in \(\Omega \). Then for a fixed great sphere \(\pi \), and denoting by \(\eta _\pi \) the vector field that is opposite to the gradient vector field of the functional \(d(., \pi )\), we have:

$$\begin{aligned} \frac{d}{dt} \log \frac{\sin d(x, \pi )}{\sin d(\alpha (t), \pi )} \ = \frac{\cos d(\alpha (t), \pi )}{\sin d(\alpha (t), \pi )} \langle \dot{\alpha }(t), \eta _\pi (\alpha (t)) \rangle \end{aligned}$$

and

$$\begin{aligned} \frac{d^2}{dt^2} \log \frac{\sin d(x, \pi )}{\sin d(\alpha (t), \pi )} \Big |_{t=0}&= \frac{1}{\sin ^2 d(y, \pi )} \langle \dot{\alpha }(0), \eta _\pi (x) \rangle ^2 \\&+ \frac{\cos d(y, \pi )}{\sin d(y, \pi )} \left( \frac{d}{dt} \langle \dot{\alpha }(t), \eta _\pi (\alpha (t)) \rangle \Big |_{t=0} \right) . \end{aligned}$$

First note that for any \(p \in \Omega \) and for any supporting hyperplane \(\pi \in \mathcal{P }\) we have \(0 < d(p, \pi ) < \pi /2\) as explained in Remark 3.10. Hence the values of \(\sin d(y, \pi )\) and \(\cos d(y, \pi )\) are strictly positive for all \(y \in \Omega \) and \(\pi \in \mathcal{P }\).

Secondly note that the quantity \(\langle \dot{\alpha }(t), \eta _\pi (\alpha (t)) \rangle \) is equal to \(\Vert \dot{\alpha }(t) \Vert \cos \theta (t)\) where \(\theta (t)\) is the angle between the velocity vector \(\dot{\alpha }(t)\) and the unit vector \(\eta _\pi (\alpha (t))\), and that the speed \(\Vert \dot{\alpha }(t) \Vert \) is constant over time. Now the positive curvature of \(S^n\) implies that the angle \(\theta (t)\) is decreasing in \(t\), which in turns implies

$$\begin{aligned} \frac{d}{dt} \langle \dot{\alpha }(t), \eta _\pi (\alpha (t)) \rangle >0. \end{aligned}$$

Therefore the function \(\displaystyle \log \frac{\sin d(x, \pi )}{\sin d(\alpha (t), \pi )}\) is convex in \(t\).

Recall that the supremum of a set of convex functions is convex. Hence, as a consequence of the variational formulation, we have:

$$\begin{aligned} F(x, y) = \sup _{\pi \in \mathcal{P }} \log \frac{\sin d(x, \pi )}{\sin d(y, \pi )}. \end{aligned}$$

Theorem 3.14

The Funk metric \(F(x, y)\) defined on a convex set \(\Omega \subset S^n\) is convex in the \(y\)-variable.

Note that the result of Theorem 3.14 contrasts with the case of hyperbolic geometry (Proposition 3.8). As we already recalled, in Euclidean geometry, the Funk metric \(F(x, y)\) is convex in \(y\) ([22]).

4 Hilbert metrics and their projective geometry

The symmetrization of the hyperbolic and spherical Funk metrics by taking the arithmetic mean provides a new set of Hilbert-type metrics, which we call the hyperbolic and spherical Hilbert metrics respectively. It is well known that the geometry of Hilbert metrics defined on convex sets in \(\mathbb{R }^n\) is very much related to the projective geometry of \(\mathbb{R }^{n+1}\). We shall see that the geometry of the hyperbolic/spherical Hilbert metrics, defined on convex sets of \(\mathbb{H }^n\) and \(S^n\) respectively, are also described in terms of projective geometry.

4.1 The Hilbert metric in \(\mathbb H ^n\)

Let \(\Omega \) be an open convex (possibly unbounded) subset of \(\mathbb H ^n\). We symmetrize the Funk metric by taking the arithmetic mean:

$$\begin{aligned} H(x, y)&= \frac{1}{2} \left( F(x, y) + F(y,x) \right) \\&= \frac{1}{2} \log \left[ \frac{\sinh d(x, b(x, y))}{\sinh d(y, b(x, y))} \cdot \frac{\sinh d(y, b(y, x))}{\sinh d(x, b(y, x))} \right] \end{aligned}$$

Definition 4.1

The metric \(H(x,y)\) is a Hilbert-type metric on \(\Omega \), and we call it the hyperbolic Hilbert metric of \(\Omega \).

The Hilbert metric satisfies the triangle inequality, obtained by adding both sides of the inequalities \(F(x, y) + F(y,z) \ge F(x,z)\) and \(F(z, y) + F(y,x) \ge F(z, x)\).

For a convex set \(\Omega \) in \(\mathbb H ^n\), the geodesic segment connecting \(x\) and \(y\) in \(\Omega \) is a Funk geodesic realizing both lengths \(F(x, y)\) and \(F(y,x)\). This implies that the geodesic segment is a Hilbert geodesic. The criterion for uniqueness of geodesics joining two points is the same as the one in the Euclidean case. More precisely, we have:

Proposition 4.2

(Hilbert geodesics) For any convex subset \(\Omega \) of \(\mathbb H ^n\), the following holds:

  1. (1)

    the hyperbolic geodesic segments are Hilbert geodesics;

  2. (2)

    the hyperbolic geodesics are the unique Hilbert geodesics joining their endpoints if and only if there does not exist in \(\partial \Omega \) two hyperbolic geodesic segments of nonempty interior which span a 2-dimensional totally geodesic subspace.

4.2 The Hilbert metric in \(S^n\)

We also consider the spherical counterpart. We define the spherical Hilbert metric \(H(x,y)\) as the arithmetic symmetrization of the spherical Funk metric. We obtain a formula which is analogous to the formula of the hyperbolic Hilbert metric, except that one replaces the sinh function with the sine function. We assume that the convex set \(\Omega \) is contained in an open hemisphere. Unlike the case of the spherical Funk metric, we do not need to assume that \(\Omega \) is contained in a sphere of diameter \(<\pi /2\), for the value \(H(x,y)\) is always nonnegative.

Definition 4.3

(The spherical Hilbert metric) For \(x\) and \(y\) in \(\Omega \), the spherical Hilbert metric is defined by the formula

$$\begin{aligned} H(x, y)&= \frac{1}{2} \left( F(x, y) + F(y,x) \right) \\&= \frac{1}{2} \log \left( \frac{\sin d(x, b(x, y))}{\sin d(y, b(x, y))} \cdot \frac{\sin d(y, b(y, x))}{\sin d(x, b(y, x))} \right) . \end{aligned}$$

Proposition 4.2 also holds in the spherical case, with spherical geodesics replacing hyperbolic geodesics in the statement.

4.3 Cross ratio on \(S^n\) and \(\mathbb{H }^n\)

Having introduced the Hilbert metrics on \(\mathbb{H }^n\) and \(S^n\), we note that the quantities inside the logarithm can be considered as non-Euclidean cross ratios and that they encode a projective geometric information, which relates the three geometries of \(\mathbb{R }^n, S^n\) and \(\mathbb H ^n\). We shall explain this briefly.

Definition 4.4

Consider a geodesic line in Euclidean, hyperbolic and spherical geometry respectively, and let \(A_1, A_2, A_3, A_4\) be four ordered pairwise distinct points on that line. We define the cross ratio \([A_1, A_2, A_3, A_4]\), in the Euclidean case, by:

$$\begin{aligned}{}[A_2, A_3, A_4, A_1]_e :=\frac{A_2 A_4}{A_3 A_4}\cdot \frac{A_3 A_1}{A_2 A_1}, \end{aligned}$$

in the hyperbolic case, by:

$$\begin{aligned}{}[A_2, A_3, A_4, A_1]_h := \frac{\sinh A_2 A_4}{\sinh A_3 A_4} \cdot \frac{\sinh A_3 A_1}{\sinh A_2 A_1}, \end{aligned}$$

and in the spherical case, by:

$$\begin{aligned}{}[A_2, A_3, A_4, A_1]_s := \frac{\sin A_2 A_4}{\sin A_3 A_4}\cdot \frac{\sin A_3 A_1}{\sin A_2 A_1}, \end{aligned}$$

where \(A_i A_j\) stands for the distance between the pair of points \(A_i\) and \(A_j\), which is equal to the length of the line segment joining them. (For this, we shall assume that in the case of spherical geometry the four points lie on a hemisphere; instead, we could work in the elliptic space, that is, the quotient of the sphere by its canonical involution.)

In the appendix, we give a proof of the fact that the quantity

$$\begin{aligned} \frac{\sinh d(x, b(x, y))}{\sinh d(y, b(x, y))} \cdot \frac{\sinh d(y, b(y, x))}{\sinh d(x, b(y, x))} \end{aligned}$$

is a projective invariant in \(\mathbb{H }^n\). However, we now give another proof of the projective invariance using the geometry of the ambient space \(\mathbb{R }^{n+1}\) for the projective model.

4.4 Perspectivities and the Hilbert metric

More details concerning this section are contained in our paper [20].

We denote by \(U^n\) the open upper hemisphere of \(S^n\) equipped with the induced metric. Let \(X\) and \(X^{\prime }\) belong to the set \(\{\mathbb{R }^n, \mathbb{H }^n, U^n\}\).

Definition 4.5

A map \(P: X\rightarrow X^{\prime }\) is a perspectivity, or a perspective-preserving transformation if it preserves geodesics and if it preserves the cross ratio of quadruples of points on geodesics.

(We note that these are classical terms, see e.g. Hadamard [12] or Busemann [7]. We also note that such maps arise indeed in perspective drawing.) The obvious examples of perspectivities are the projective transformations of \(\mathbb{R }^n\) to itself. In what follows, using well-known projective models in \(\mathbb R ^{n+1}\) of hyperbolic space \(\mathbb H ^n\) and of the sphere \(S^n\), we define natural homeomorphisms between \(\mathbb R ^n, \mathbb H ^n\) and the open upper hemisphere of \(S^n\) which are perspective-preserving transformations.

In fact, the three cross-ratios are the manifestation of the same entity; they are obtained from each other via projection maps between familiar representatives of the three geometries in \(\mathbb R ^{n+1}\). More specifically, the sphere \(S^n\) is the set of unit vectors in \(\mathbb{R }^{n+1}\) with respect to the Euclidean norm

$$\begin{aligned} \Vert x\Vert _e^2:=x_1^2 + \cdots + x_n^2 + x_{n+1}^2 = 1 \end{aligned}$$

and the hyperbolic space \(\mathbb{H }^n\) is the set of “vectors of imaginary norm \(i\)” with \(x_{n+1}>0\) in \(\mathbb{R }^{n+1}\) with respect to the Minkowski norm

$$\begin{aligned} \Vert x\Vert _m^2:= x_1^2 + \cdots + x_n^2 - x_{n+1}^2 = -1. \end{aligned}$$

These models of the two constant curvature spaces are called “projective” for the geodesics in the curved spaces are realized as the intersection of the unit sphere with the two-dimensional subspace of \(\mathbb{R }^{n+1}\) through the origin of this space.

Let \(P_s\) be the projection map from the origin of \( \mathbb{R }^{n+1}\) sending the hyperplane \(\{ x_{n+1} = 1 \} \subset \mathbb{R }^{n+1}\) onto the open upper hemisphere \(U^n\) of \(S^n\).

Let \(P_h\) be the projection map from the origin of \( \mathbb{R }^{n+1}\) of the unit disc of the hyperplane \(\{ x_{n+1} = 1 \} \subset \mathbb{R }^{n+1}\) onto the hyperboloid \(\mathbb{H }^n \subset \mathbb{R }^{n+1}\).

Theorem 4.6

(Spherical Case) The map \(P_s\) is a perspectivity. In particular, the projection map \(P_s\) preserves the values of the cross ratio; namely for a set of four ordered pairwise distinct points \(A_1, A_2, A_3, A_4\) aligned on a great circle in the upper hemisphere, we have

$$\begin{aligned}{}[P_s(A_2),P_2(A_3),P_s(A_4),P_s(A_1)]_s=[A_2, A_3, A_4, A_1]_e. \end{aligned}$$

As the proof is elementary, and as it is short, we include it.

Proof

Let \(u, v\) be the two points on the hyperplane \(\{x_{n+1} =1\}\) and \(P_s(u)=:[u], P_s(v)=:[v]\) be the points in \(U\), and \(d([u], [v])\) be the spherical distance between them. Let \(\Vert x\Vert \) be the Euclidean norm of the vector \(x \in \mathbb{R }^{n+1}\). We show that

$$\begin{aligned} \sin d( [u], [v] ) = \frac{\Vert u-v\Vert }{\Vert u\Vert \Vert v\Vert }. \end{aligned}$$

This follows from the following trigonometric relations:

$$\begin{aligned} \sin d( [u], [v] )&= \sin \left[ \cos ^{-1} \left( \frac{u}{\Vert u\Vert } \cdot \frac{v}{\Vert v\Vert } \right) \right] \\&= \sqrt{1 - \cos ^2 \left[ \cos ^{-1} \left( \frac{u}{\Vert u\Vert } \cdot \frac{v}{\Vert v\Vert } \right) \right] } \\&= \sqrt{1- \left( \frac{u}{\Vert u\Vert } \cdot \frac{v}{\Vert v\Vert } \right) ^2}\\&= \frac{1}{\Vert u\Vert \Vert v\Vert } \sqrt{\Vert u\Vert ^2 \Vert v\Vert ^2 - (u \cdot v)^2} \\&= \frac{1}{\Vert u\Vert \Vert v\Vert } \times ( \text{ area } \text{ of } \text{ parallelogram } \text{ spanned } \text{ by } u \text{ and } v) \\&= \frac{\Vert u-v\Vert }{\Vert u\Vert \Vert v\Vert }. \end{aligned}$$

Now for a set of four ordered pairwise distinct points \(A_1, A_2, A_3, A_4\) aligned on a great circle in the upper hemisphere, their spherical cross ratio \([A_2, A_3, A_4, A_1]_e\) is equal to the Euclidean cross ratio \([P_s(A_2), P_s(A_3), P_s(A_4), P_s(A_1)]_s\);

$$\begin{aligned} \frac{ \sin d( [A_2], [A_4] )}{\sin d([A_3], [A_4])} \cdot \frac{ \sin d( [A_3], [A_1] )}{\sin d([A_2], [A_1])}&= \frac{\frac{\Vert A_2-A_4\Vert }{\Vert A_2\Vert \Vert A_4\Vert }}{ \frac{\Vert A_3-A_4\Vert }{\Vert A_3\Vert \Vert A_4\Vert } } \cdot \frac{\frac{\Vert A_3-A_1\Vert }{\Vert A_3\Vert \Vert A_1\Vert }}{ \frac{\Vert A_2-A_1\Vert }{\Vert A_2\Vert \Vert A_1\Vert } } \nonumber \\&= \frac{\Vert A_2-A_4\Vert }{\Vert A_3 - A_4\Vert } \cdot \frac{\Vert A_3-A_1\Vert }{\Vert A_2 - A_1\Vert }. \end{aligned}$$

\(\square \)

Theorem 4.7

(Hyperbolic Case) The map \(P_h\) is a perspectivity. In particular, the projection map \(P_h\) preserves cross ratios, namely for a set of four ordered pairwise distinct points \(A_1, A_2, A_3, A_4\) aligned on a geodesic in the upper hyperboloid, we have

$$\begin{aligned}{}[P_h(A_2),P_h(A_3),P_h(A_4),P_h(A_1)]_h=[A_2, A_3, A_4, A_1]_e. \end{aligned}$$

The proof is given in [20]. We do not include it here because it is analogous to the proof of Theorem 4.6, replacing the computation on the sphere of unit radius in \(\mathbb{R }^{n+1}\) by a computation on the upper sheet of the sphere of radius \(i\), namely the hyperboloid in \(\mathbb{R }^{n, 1}\).

The projectivity invariance for the cross ratio in Euclidean space is now extended to an analogous property in the spherical and hyperbolic spaces (using the same definition of projectivity). In the case of the sphere, we restrict to configurations where all the points considered are contained in an open hemisphere. From this we have easily the following result (that can be traced back to old work of Menelaus in the spherical case):

Corollary 4.8

The spherical and hyperbolic cross ratios are projectivity invariants.

This follows from the facts that the projection map \(P_s\) and \(P_h\) are both perspective-preserving transformations, and that cross ratio in the Euclidean space is a projectivity invariant.

By translating these results in the language of Hilbert geometry, we have:

Corollary 4.9

(Spherical Case) Consider an open convex set \(\Omega \) in \(S^n \subset \mathbb{R }^{n+1}\) contained in the upper hemisphere \(\{\Vert x\Vert _e^2 =1, x_{n+1}>0\}\), and let \(H\) be the spherical Hilbert metric of \(\Omega \). Let \(\tilde{\Omega }\) be the image of \(\Omega \) by the map \(P_s^{-1}\) in the hyperplane \(\{ x_{n+1} = 1 \}\), and let us equip \(\tilde{\Omega }\) with its Euclidean Hilbert metric \(\tilde{H}\). Then the map \(P_s^{-1}: (\Omega , H) \rightarrow (\tilde{\Omega }, \tilde{H})\) is an isometry.

Corollary 4.10

(Hyperbolic Case) Consider an open convex set \(\Omega \) in \(\mathbb{H }^n \subset \mathbb{R }^{n, 1}\) where the inclusion is the isometric embedding of \(\mathbb{H }^n\) as the hyperboloid \(\{ \Vert x\Vert _m^2 = -1 , x_{0} >0\}\), and let \(H\) be the hyperbolic Hilbert metric of \(\Omega \). Let \(\tilde{\Omega }\) be the image of \(\Omega \) by the map \(P_h^{-1}\) in the hyperplane \(\{ x_{0} = 1 \}\), and let us equip \(\tilde{\Omega }\) with its Euclidean Hilbert metric \(\tilde{H}\). Then the map \(P_h^{-1}: (\Omega , H) \rightarrow (\tilde{\Omega }, \tilde{H})\) is an isometry.

Note that there are no analogues of Corollaries 4.9 and 4.10 in the Funk setting.

Denote by \(\mathbb D _R=\mathbb D ^n_R \subset \{x_{n+1} = 1\}\) a disk of radius \(R>0\) centered at the origin of Euclidean \(n\)-space. By restricting the projection \(P_s\) to the disk \(\mathbb D _R\), we obtain a map from this disk into the sphere \(S^n\). For \(0<R < 1\), by restricting the projection map \(P_h\) to \(\mathbb D _R\), we obtain a map from this disk into the hyperboloid \(\mathbb H ^n\). Since these maps are isometries for the Hilbert metrics and since the disk \(\mathbb D ^n_R\) equipped with its Hilbert metric is a model of hyperbolic geometry (the Beltrami–Klein model), we obtain models of hyperbolic space which sit in hyperbolic space and in the sphere respectively. We call such models generalized Beltrami–Klein models, since they are defined using the spherical and the hyperbolic cross ratios respectively. Thus we identify, in each of the positively/negatively curved spaces, a nested family of geodesic balls as models of the hyperbolic space. Note that the limit of these models in \(S^n\) and in \(\mathbb H ^n\) as the radius \(R\) goes to \(\pi /2\) and \(\infty \) respectively, is the upper cap \(U^n\) and the entire space \(\mathbb H ^n\) whose Hilbert metrics, if we define them by the formula we used for proper open convex subsets, are identically zero.