On the entropy of Hilbert geometries of low regularities

We compare the regularity of the boundary of a convex set with the value of its Finslerian volume entropy. The main result states that the volume entropy of a two-dimensional domain whose associated curvature measure is Ahlfors α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-regular is 2αα+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{2\alpha }{\alpha +1}$$\end{document}.


Hilbert geometries
To define a Hilbert geometry, we need a convex compact subset of R n (or a strictly convex set in RP n ).Then we construct a distance in the interior of the convex set using the cross-ratio (Fig. 1).
Precisely, take two points p and q in Int( ).The compactness and convexity show that there exist two uniquely determined points a and b on ∂ such that a, p, q, b are aligned in this order.We set When is an ellipsoid, we construct the Klein model for hyperbolic geometry.For any other case, the distance is not even Riemannian [13].However it is Finslerian, infinitesimally where t 1 and t 2 are positive real numbers such that x + t 1 v and x − t 2 v meet the boundary of (Fig. 2).The couple ( , d ) is called a Hilbert geometry.Such spaces enjoy the following remarkable properties [19]: • The metric spaces ( , d ) are complete.In particular, the boundary ∂ is metrically at infinity.• (Affine or projective) straight lines are geodesics.
• The group of projective transformations leaving invariant acts by isometries on ( , d ).
Remark 1 Throughout this text, we will use the notation |•| for refering to Euclidean lengths, norms or volumes and • will be used for the Finslerian quantities.
Finsler balls are denoted B(x, R).Euclidean balls are denoted B euc (x, R) and balls for an arbitrary metric space (X , d) are denoted B d (x, R).We use similar conventions for spheres

Entropy
In this context of Hilbert geometries, the main goal of this article is to study an invariant, the volume growth entropy.The volume entropy of a metric measured space (X , d, μ) is the exponential asymptotic growth rate of volume of balls when the radius goes to infinity.
Precisely, it is the real number defined as the limit (whenever it exists) It is known to be a powerful invariant.When (X , d) is a Riemannian manifold and μ is the Riemannian volume, it has been used several times to capture a lot of informations about the ambient geometry (see for instance [4]).
Returning to the context of a Hilbert geometry ( , d ) given by a convex ⊂ R n , one of the common issues is that the volume is not canonically defined.However we can isolate axioms for a notion of appropriate volume ( [2]) for which the entropy does not depend on the choice of an appropriate volume.In this paper, we compute volumes with the so-called Hausdorff (n-)measure.Let us define it.
We consider the function σ on given by where ω n is the measure of the unit Euclidean ball of R n , L is the Lebesgue measure and B(x, 1) is the Finslerian unit ball.Finally the Hausdorff measure μ is the measure (absolutely continuous with respect to the Lebesgue measure), the density of which is given by σ , i.e: for any Borel set A. The density σ is called the Busemann function.The fact that μ is indeed the n-dimensional Hausdorff measure follows from [6].
For a general Hilbert geometry, the quantity log(μ(B(x, R))) R does not converge in general when R goes to infinity ( [21] corollary 4), so we usually consider lower and upper entropies: For instance, when is an ellipsoid in R n , The notation h implicitly means that log(μ(B(x, R))) R converges and is independent of x.The volume entropy of Hilbert geometries has been studied by various authors.For instance it has been proved by Tholozan in [20] that the entropy of Hilbert geometries never exceed the hyperbolic entropy.This result has been known in dimension 2 and 3 since the work of C. Vernicos in [21] and in the case of divisible convex sets thanks to a result of M. Crampon in [8].It is known that the entropy can take any value in dimension 2 ( [21] corollary 4) and it has been precisely computed in many cases: for instance the entropy vanishes for convex polytopes [10] and it is extremal as soon as the convex set is sufficiently regular.Actually the latter statement is the starting point of this paper, let us make it precise: Theorem 2 (First main theorem in [3]) Suppose the boundary of the convex set is a hypersurface of R n of regularity C 1,1 .Then the entropy exists and h( ) = n − 1.

1.3 Question
The question we would like to address in this paper is suggested by the previous result and is the following Question 3 Can we find a relation between the regularity of the boundary of a convex domain and the value of its Finslerian volume entropy ?
In particular, is there a lower bound for the entropy of a domain given the regularity of its boundary?And are there lower regularities for the boundary than C 1,1 which guarantee maximal entropy?

Results
We propose two types of answer for the previous question.We first describe a bijective relation between entropy and regularity in dimension 2. Following Theorem 2, a natural (but slightly naive) approach would be to try to understand a C 1,α regular convex set, for α < 1.As a global invariant, the volume entropy only sees the part of the convex set with maximal volume growth.In particular, as soon as there is a part of the boundary of which has regularity C 2 with positive Gauss curvature, then h( ) = n − 1.This may happen for a C 1,α regular convex set and this implies that in particular there is no hope of characterizing the entropy of a domain with its regularity C 1,α .Hence, in order to precisely compare the regularity and the entropy, we need the regularity to realize the following constraint: "Whenever the Gauss curvature is positive (in a possibly weak sense), then the regularity must be strictly C 1,α but not more."This is roughly the definition of Ahlfors α-regularity.We postpone until Sect. 2 the precise definition of Ahlfors regularity.The main result of this paper is then the following.
Theorem 4 (First main theorem) Let be a convex and relatively compact domain of R 2 which is Ahlfors α-regular.Then h( ) = 2α α + 1 .

Remark 5
The limit set of a convex-cocompact Fuchsian group is a Ahlfors α-regular Cantor set for some α < 1 (in dimension 2).Let be the convex hull of this limit set in H 2 .By a result of Patterson [18], we know that the hyperbolic exponential growth inside is exactly α.In Theorem 4, the growth is faster since 2α α+1 > α and this reflects the fact that the Hilbert distance in is always bigger than the (trace of the) hyperbolic distance.Indeed, when approaching an edge of which is inside H 2 , the Hilbert metric blows up, whereas the hyperbolic metric does not.
In a second and independent part of this paper, we show a stronger version of Theorem 2, weakening the assumption of C 1,1 -regularity.The space of C 1,1 maps is isomorphic to the Sobolev space W 2,∞ .We show that for some finite values of p, domains whose boundaries are W 2, p -regular must have volume entropy equal to n − 1. i.e. we show that we can decrease the Sobolev regularity and still be sure to get a convex set of maximal volume entropy.
Theorem 6 (Second main theorem) Let be a convex relatively compact subset of R n .Assume that the boundary is parametrized by a homeomorphic map ϕ : The number C(x, ) already appears in [3] and is denoted there as the centro-projective area (see Paragraph 4.2).For the moment, the reader may think of the centro-projective area as the total curvature of the boundary.
Note that the behavior of entropy for W 2, p boundaries for p < n − 1 is still completely open.From the Sobolev embedding theorem, if p > n − 1, the assumption on C 1 -regularity is superfluous.

Outline of the proof and plan of the paper
The proofs of Theorems 4 and 6 are completely independent but both have an intense flavor of geometric measure theory.
Sections 2 and 3 are devoted to Ahlfors regularity and the proof of Theorem 4, while Sects.4 and 5 deal with Sobolev regularity and the proof of Theorem 6.
The proof of the first main theorem follows two steps.First, we define and compute the volume entropy of some reference Ahlfors α-regular convex sets, called Cantor-Lebesgue convex sets and constructed with the familiar Cantor-Lebesgue "the devil's staircase" map.Those convex sets are defined in Sect. 2 and the entropy computation is achieved in Paragraph 3.1.The second stage of the proof is a comparison argument: taking an arbitrary Ahlfors α-regular convex set, we show that its volume entropy is the same as the entropy of the Cantor-Lebesgue convex set of same regularity.Ahlfors α-regular maps have a welldefined weak second derivative, which is a measure supported on a Cantor set.In order to compare the volume entropies of the convex domains, we first need to compare the associated Cantor sets.To this end, we use and generalize the main theorem of [16] in Paragraph 3.2 and we show that the Cantor sets can be related by an order preserving bi-Lipschitz map.The conclusion follows in Paragraph 3.3.
The second theorem is very close to the main theorem of [3], the novelty here is concentrated in Lemma 30.Consequently we follow the same outline for the proof.The first step is to show that the (suitably renormalized) Busemann function σ ( p) converges to the curvature of the boundary as p approaches the boundary.As in [3], the main argument lies in the work of D. Alexandroff [1].This convergence is pointwise almost everywhere in [3] and L 1 in this paper (we chose p ≥ n − 1 so that this L 1 convergence makes sense).We then apply Lebesgue Dominated Convergence theorem to deduce that the renormalized volume of balls converges to the centro-projective area.The key Lemma 30 is specific to the Sobolev regularity context and guarantees the conditions of the Dominated Convergence Theorem.

Ahlfors regularity
Definition 1 (1) Let (X , d, μ) be a metric measure space.We say that it is Ahlfors α-regular if there exists a constant C > 0 such that, for any x ∈ X and any r > 0, (2) Let ϕ : [0, 1] → R be a non-decreasing continuous map and let μ be its derivative in the sense of distributions, seen as a measure.We denote by X the support of this measure. We Lemma 7 Let ϕ : [0, 1] → R be a continuous function, μ its derivative and X the support of μ.The function ϕ is Ahlfors α-regular if and only if, there exists C > 0, such that for any s, t ∈ X with ϕ(s) = ϕ(t), we have This lemma explains why we claimed in the introduction that α-Ahlfors regular maps are thought as maps of regularity C α which are not more regular whenever the second derivative is positive.
Proof This follows from the very definition of the derivated measure: Throughout the rest of this section we fix a parameter p ∈ ]2, +∞[.We now describe the construction and the useful properties of the Cantor-Lebesgue domains p .In particular we show that the weak second derivative is supported on a log 2 log p -regular set.

Construction of Cantor-Lebesgue domains
The Cantor-Lebesgue map is constructed inductively.We start with the identity function f 0 : x → x, then inductively define (Fig. 3) The following facts are well known: The sequence ( f n ) converges uniformly to a non-decreasing continuous map f .(2) The limit map f is log 2 log p -Hölder continuous.
Fig. 3 The sequence is clearly satisfied.Hence ( f n ) form a uniform Cauchy sequence of continuous maps, converging to some function f .Now suppose The biggest gap of two points separated by distance And finally

A probability measure on the Cantor set
The function f is continous and, as such, it has a weak derivative.It is a Borel measure μ given by Since ( f n ) converges uniformly, we can also view μ as the limit of the sequence (μ n ) of derivatives of the f n 's.The supports of the measures μ n are decreasing.Then is a Cantor set.We denote this Cantor set by C p .

Lemma 9
The measure μ is log 2 log p -Ahlfors regular.
Proof By virtue of being Hölder continuous, we have the upper bound for Ahlfors regularity.
To prove the lower bound, let x ∈ C p ; it can be written as the countable intersection of closed intervals {x} = n I n,k n of size p −n .By construction the measure μ applied to any Cantor interval of size p −n yields 2 −n .Hence for 2 p −n ≤ r < 2 p 1−n we have With Lemma 7 above, this also shows that f is log 2 log p -Ahlfors regular.
Now we can explicitly construct the domains.Consider the map e iπ f .Its values give unit vectors in R 2 .If we define γ (t) = t 0 e iπ f , this a curve.If we take a second copy and rotate by π around the point (γ (0) + γ (1))/2, we obtain the boundary of a set, which is convex by the monotonicity of f .This explicitly defines a map and extending periodically.This set is the Cantor-Lebesgue domain.Throughout the rest of this text, we denote this set by p .Definition 2 Let be a convex relatively compact set of R 2 .We say that (or sometimes ∂ ) is Ahlfors α-regular if ∂ could be written locally as the graph of a C 1 function ϕ : R → R whose derivative is an Ahlfors α-regular map.
Note that, in this case, the curvature measure of (see the definition at the beggining of Paragraph 4.2) is an Ahlfors α-regular measure supported on a Cantor set of dimension α.Be aware that the definition of Ahlfors α-regularity refers to the weak second derivative and not the first.However, the notation Ahlfors 1 + α-regular seems too much misleading.

Theorem 10 For every p ∈]2, ∞[ the Cantor Lebesgue domain p has entropy
where α = log 2/ log p Remark 11 By theorem 2.14 in [3], as a corollary of the co-area inequalities, the authors show that the entropy is also given by the asymptotic exponential growth rate of volume of spheres.Hence, from now on, we focus on computing the length of the circles of radius R and the exponential growth of this length will give the entropy.In fact by Lemma 3 in [9] we can consider the length of the scaled boundary η∂ (η < 1), which we do.

Lemma 12
Let ⊂ R 2 be a convex domain containing the origin.Then where F + (θ, η) is the point on ∂ given by the intersection of the tangent line at ηγ (θ) of η∂ and ∂ , in the positive orientation, and F − is that in the negative orientation.
Proof This is just a direct calculation of the length with the Hilbert norm of the curve which is given by We will assume that our map γ : S 1 euc → ∂ has unit speed.Let ϕ(θ) denote the generating function; so that: Lemma 13 Let ⊂ R 2 be a convex domain whose boundary generating function ϕ is given by the cumulative distribution function of an Ahlfors α-regular measure for some 0 < α < 1, and let X denote the support of this measure, then there is a number c such that for every θ as η → 1.

Convention 14 In the following proof and throughout the rest of this text, for a convex set
, and a point x ∈ ∂ , ν(x) denotes the unit inner normal of and based at x.

Proof
Let η be fixed once and for all.For each θ we want to find the point forward F + (θ, η) belonging to both the tangent line at ηγ (θ) of η and the boundary of the convex set .Hence

Now we use the Taylor approximation to sin to yield
The generating function ϕ is the cumulative distribution function of an Ahlfors α-regular measure on X , where α = log 2/ log p. Consequently, there is a number c 1 ≥ 0 such that θ ∈ [0, 2π]\X , and 0 ≤ t ≤ 1/2 we have ), 0} α Now we apply this to the previous bound to get for h < dist + (θ, X ) that the integral is zero, and for h ≥ dist + (θ, X )

Now if we can bound h such that
but by virtue of convexity the scalar product to the normal to ∂ with the position is bounded above and below away from 0: there is a number c 3 such that for every θ ∈ [0, 2π] , and so and If we invert everything, and note that The previous lemma applies whether θ belongs to the support X of the measure or not; but in the following proof, we may only apply it for θ / ∈ X .We note that a similar result holds for F − (θ, η).

Proof of theorem 10
Now we take advantage of the fact that the standard measure on the Cantor-Lebesgue set is Ahlfors α-regular for α = log(2)/ log( p).We consider a division of the complement of X into a union of intervals of progressively smaller sets.At generation N the size of the sets is ( p −2) p −N , and there are 2 N of them.Now we note that for an antipodal set B(0, R) its boundary given by tanh Because all terms uniformly constant in R will converge to 1 as we take the 1/R power, it is suffiction to consider

Now we can break this sum into the generations to yield
p N  .
In order to proceed we will have to estimate the sum by splitting it in two.Let We investigate separately the sum over indices N where N < 2Rβ and where and the two split sums will have very different asymptotic behaviour.
We estimate the first term below coarsely And from we use the fact that if x ≥ 1, then log(1 + cx) ≤ 1 + log x + max(0, log(x)): Now for the second term, we first use the approximation log(1 + x) ≤ x for any x.We have For the lower bound, we want to use and x/2 ≤ log(1 + x) for 0 < x < 1.If 2Rβ is an integer, the first term of the sum is log(1 + ( p − 2)) and p − 2 is not necessarily smaller than 1.To avoid this situation we split again the sum into two parts and take off the two first terms.
N ≥2Rβ The very same argument applies for F − instead of F + .Consequently we have a two-side bound: and If we take the R th root and let R go to infinity we arrive at lim Taking the logarithm yields the result.
We now come to the general case of a general Ahlfors α-regular convex set (recall Definition 2).The aim of the rest of this section is to prove the first main theorem, Theorem 15 Let be a 2-dimensional Ahlfors α-regular convex set.Then its volume entropy satisfies h( ) = 2α α + 1 .

Comparison of Ahlfors regular sets
In this section, we compare Ahlfors regular Cantor sets, as subsets of R.
Remark 16 In this paragraph, we argue with real Ahlfors regular sets.This corresponds to spherical Ahlfors regular sets with the stereographic projection.Since Ahlfors regularity is defined locally, an Ahlfors regular domain has the same volume growth as the domain minus a "slice" for any unit vector v (we assume that o ∈ R 2 is inside ).
On the boundary of the later set, stereographic projection is a bilipschitz map.
The Ahlfors regular sets that we considered in this section are then stereographic projections of the boundary of minus a slice.We come back to spherical sets in the next section.
Let us start by fixing notations.We denote by κ an Ahlfors s-regular measure supported on an Ahlfors s-regular set E. In the next paragraph we will apply our result below where κ is the curvature measure of a α-regular Cantor set.
We denote a standard Cantor set (instead of Cantor-Lebesgue Cantor set) by C t .By definition, such a Cantor set is obtained as the support of the derivative of a Cantor-Lebesgue function which is Ahlfors t-regular.
Theorem 17 (ordered s-regular embeddings) Let E be a totally ordered s regular set.Let F be a totally ordered t regular set.Assume one of the two regular sets is a standard Cantor set.If s < t, there is an order preserving bi-Lipschitz embedding ϕ : E → F whose constants depend only on the diameters of E and F, their Ahlfors constants, and t and s.

Remark 18
This statement is an extension of the theorem 3.3 in [16].Here we add the fact that we can choose the bi-Lipschitz map ϕ to be order-preserving.Note that our proof uses the result of [16].
Starting with the map f : E → F given by theorem 3.3 of [16], we proceed as follows Step one: construction of the binary tree Let E be an ordered Ahlfors s-regular set.As being closed, we can write the complement of E as a disjoint union of open intervals: On the Set A 0 = [0, 1] and, for k > 0, We build a finite tree for which the vertices are the elements of V n .We now describe the set of edges.We first remark that the union defining V n is not disjoint: passing from A k to A k+1 creates a hole in one of the v k i .So precisely there are in v k+1 0 , . . ., v k+1 k+1 exactly 2 new elements and there is precisely one element in and the union is disjoint.For each k, we place two edges: We get a finite binary tree T n .Finally we consider the R-tree T = ∞ n=0 T n .If needed we put the Cantor set in the notation and denote the tree by T E .

Claim Every vertex has exactly two descendants.
Proof From the construction, it is obvious that each vertex has either 0 or 2 descendants.Suppose there exists v k i with no descendant.Then v k i ⊂ E and E is not totally discontinuous; a contradiction.
Let us now endow the binary tree T with a metric.Given a vertices v k i and v l m , we define the distance between them to be the size |v| of the smallest closed interval which contains both of them.This metric is incomplete, the vertices accumulate on the boundary which is at finite distance (one) to the basepoint v 0 0 .We denote by d T the metric on ∂ T induced by the completion of the metric described above.
Step two: comparison of metrics on the regular set We have two metrics on the boundary ∂ T = E: the tree metric d T and the Euclidean metric d = |•|, coming from the inclusion E ⊂ R and we now compare those two metrics.

Theorem 19 The tree metric d T and Euclidean metric d are bi-Lipschitz equivalent on E.
Proof The proof will proceed in several parts.First we will show that Ahflors regularity implies that for a given vertex v in our tree with descendents u and w, it follows that there is a number K > 0 which depends only on s and the proportionality constants of s-regularity, such that As a result we can deduce that there exist c and C such that From this it follows that where K dependts only on s, c and C. Note that |I | = |v| − |u| − |w| so that we got the first part of our goal inequality.Conversely, we want to show that any descendant u or w has a Euclidean length bounded by K times the Euclidean size of the gap |I |.To achieve that, we cover v in balls of radius |I | and centered in E ∩ v.This is possible, because |I | is the largest gap in v ∩ E. First we have a finite sub-cover.We can then take a Vitali cover to yield balls where K and C depend only on s,c and C.
It is now clear how we can show that d and d T are bi-Lipschitz equivalent.Indeed, let x, y ∈ E and let v be the smallest closed interval containing both ot x and y and w be the biggest open interval contained in [x, y].We have Step three: extension of f We are guaranteed a bi-Lipschitz map from E to C t for any standard Cantor set t > s by Theorem 3.3 in [16].Let f denote this function.Let f : T E → T C t be given by Let I ⊂ f (v) be the largest open interval (with possible semantic order) contained in f (v).By construction there are an x, y ∈ v ∩ E such that f (x) and f (y) are separated by I , so For the reverse inequality, note that we have a K depending only on t, such that f (u ) and f (u ) are separated by a set of size at least (K ) −1 d T ( f (u ), f (u )).Consequently there is an x ∈ u and a y in u such that Step four: reordering Subsequently, the most useful property of the standard Cantor set is its self similarity.In particular for the standard Cantor tree T C t with the metric d T , if we flip any branch, it is an isometry of the tree T C t , as is any uniform limit of isometries.Because our tree is binary we can identify each vertex with a word composed of the letters l and r , for left and right.Now given u and v which are not descendents we say that u is to the left of v if their minimal closed cover is w is such that u is in the left branch from w and v is in the right branch.We proceed inductively on word length.Suppose φ n • f preserves the order of the first vertices with word length less than or equal to n.For each vertex w of word length n consider its two descendents u and v such that u is to the left of v.If φ n • f (u) is to the left of φ n • f (v) proceed if not then postcompose φ n with the branch flip, which flips the tree at the common vertex of φ n • f (u) and φ n • f (v).This flip leaves every other descendent pair unchanged because this common vertex is a descendent of φ n • f (w).Let φ n+1 be the end result.
In this way we construct a sequence of isometries which converges uniformly to a limit isometry φ.Then f = φ • f is a bi-Lipschitz map which preserves the semantic order.In fact we get the following Corollary 20 Let E and E be s and t regular subsets of [0, 1], one of which is the standard regular set C t or C s .Suppose {0, 1} ⊂ E. There is an order-preserving map f : [0, 1] → [0, 1] which is bi-Lipschitz on its image, and such that f (E) ⊂ E .
Proof We take a standard Cantor set C with s < τ < t.Then there is an order preserving bi-Lipschitz map f : E → C and g : C → E .First we extend f to [0, 1].We do this by mapping each open interval I on the complement to the image of it's endpoints and scaling linearly.Because the map f is bi-Lipschitz and order preserving, so is the extended map.By composing the extensions we get the desired map.

Entropy of a domain with curvature an Ahlfors regular measure
Recall that we dispose of a 2-dimensional convex set , which is Ahlfors α-regular: its curvature measure is supported on E, an α-regular Cantor set.We choose t > α and we let C t denote the standard Cantor set.
As in Remark 16, we cut out a slice of (which does not change the volume growth) so that we can bilipschitzly map E to an α-regular subset of R.
We can express the complement of the support of our measure as a union of open intervals where I j =]a j , b j [, with a j , b j ∈ E. We know from the previous paragraph that, associated to C t , we have a bi-Lipschitz order preserving map ϕ : E → C t .As in the case of standard Cantor set, the lengths of the Finslerian circles of radius R in are given by a sum of integrals of the form x + e −2R/(α+1) dx.
where I are the open intervals whose disjoint union makes the complement of the t dimensional cantor set.This last inequality follows from the elementary identity log(1 whenever λ and μ are positive.Because our map is order preserving, we can just estimate from above (there will be no open intervals appearing more than once in the image) Here p is the real number such that log 2 log p = t = dim C t .Then we proceed as before.Setting β = 1 (α+1) log p , we break the sum into two parts where either n < 2Rβ or n ≥ 2Rβ and use the same techniques for bounding above: using for the first term the identity log(1 + x) ≤ 1 + log x and for the second log(1 which we can coarsely bound by 2 2Rβ +1 2Rβ log p. For the second term (n ≥ 2Rβ), we have Consolidating terms we get that the integral is bounded by Noting log 2β = t/(α + 1) and taking the power of 1/R, the logarithm and letting R go to infinity yields the entropy bound Ent(E) ≤ 2t/(α + 1).
But t is arbitrarily close to α.We obtain the lower bound in the very same way: we use this time an order preserving map ψ : C s → E and produce a lower bound with the same techniques as for the standard Cantor set.

What is a Sobolev-regular convex set?
We recall that we denoted by ν(x) the unit inner normal, for a point x ∈ ∂ .
In this section, we clarify the definition of W 2, p convex domains (via charts) and show that the definition is equivalent to requiring that the boundary can be parametrized by a map ), which we define to be the space of continuously differentiable functions f : We say a domain has regularity W 2, p ∩ C 1 for p ≥ (n − 1) if for every x ∈ ∂ there is an n − 1 dimensional supporting plane P through x, and open subsets U of P and U of ∂ containing x such that U is given by the graph of a function f ∈ W 2, p ∩ C 1 (U ) i.e. for every x there is a U ⊂ T x ∂ containing 0, and a function h Note that for p = n − 1, W 2, p (S n−1 euc ) does not embed in C 1 which is why we study parametrizations in W 2, p ∩ C 1 .If p > n − 1, W 2, p (S n−1 euc ) does embed in C 1 .We define the barycenter of a domain as the center of mass of the boundary bc( Projecting the unit sphere around the barycenter onto ∂ gives a map.ϕ : S n−1 euc → ∂ which takes θ ∈ S n−1 to the point on ∂ in the direction θ from bc( ) If is a bounded convex domain then bc( ) exists as does ϕ .

regular if and only if the map
which takes z to ∂ along ν(x), and then radially to the unit sphere centered at x 0 .This map is C 1 , as it is a quotient of C 1 functions.It is also Consider the differential of at z, Notice that this is the composition of the linear map between the two tangent hyperplanes which is injective hence invertible, and the orthogonal projection from which is also invertible because the line from where (D F) −1 is the matrix inverse of D F. This is a composition of continuous functions, so is continuous.Now taking the derivative with respect to a vector where we have used the fact for any invertible matrix valued function The change of variables is at least a C 1 diffeomorphism and the matrices D F and D F −1 are continuous (and hence bounded).This imply that D X D(F −1 )(x) is bounded uniformly almost everywhere by D F ∞ D F −1 ∞ |D 2 F|.This is valid for any C 2 approximation of F, and so passing to the limit is valid for F [11,Chap. 5].And so if For the reverse implication, let x be a point in ∂ .Then consider the map = T x ∂ •ϕ where T x ∂ is the orthogonal projection onto the tangent plane at x of ∂ .By a similar chain of reasoning is in W 2, p ∩ C 1 (S n−1 euc , T x ∂ ) and is locally invertible around ϕ −1 (x).By a similar argument to the forward implication, it too is in W 2, p ∩ C 1 .Then h(z) is given by (ϕ • −1 )(z), ν(x) which is a composition of W 2, p ∩C 1 functions which is W 2, p ∩C 1 , [17].

The curvature measure and the centro-projective area for the Sobolev regularity
Let be a convex compact set.Convex domains naturally have some regularity.They are automatically Lipschitz regular, and twice differentiable almost everywhere but in addition we can define a set valued Gauss map for the boundary of any domain.For every x ∈ ∂ and ν ∈ S n−1 , we consider the halfspace and then we set This allows us to define a Gauss curvature measure on ∂ by where μ is the usual measure on the sphere.For a C 2 domain the measure corresponds with the change of variables formula Already the C 2 assumption implies that the entropy of a domain is n−1.In [3] this supposition was weakened to C 1,1 .If one weakens the asumption to W 2, p for p ≥ n − 1, we still have a well defined Gauss curvature.Indeed, the determinant of the Gauss map det DG belongs to L 1 because p ≥ n − 1 > n−1 2 (see the proof below).Assume now that the origin o of R n belongs to (this is no restriction) and let a : ∂ → R be the positive function such −a( p) p ∈ ∂ (see the introduction of [3]).For instance if o is a center of symmetry of , then a is just the constant 1.The letter a stands for antipodal function.We can now recall the definition of the centro-projective area in the following way (the original definition of [3] does not assume any regularity of the convex domain).Definition 3 (centro-projective area) Let be a convex set such that ∂ is parametrized by a map in W 2, p ∩ C 1 (S n−1 ).The centro-projective area is defined as Lemma 22 Let be a convex domain of class C 1 ∩ W 2, p for p ≥ n − 1.Then the centroprojective area is nonzero Proof By convexity G( p), p ≥ δ > 0 for all p.As such the angular distance between G( p) and p is uniformly bounded between 0, π/4 − δ .Assuming G is in W 1, p ∩ C 0 , we may consider a sequence G ε of smooth approximations to G converging in W 1,n ∩ C 0 .As such, eventually as ε → 0, d S n−1 ( p, G ε ( p)) ≤ π/4 and G ε is homotopic to the identity map, consequently where s is the regular volume form on S n−1 .As such, because G ε converges in W 1,n to G ε , and Thus det DG is nonzero on a set of positive measure.Furthermore, by convexity and Alexandrov's theorem on twice differentiability det DG is non-negative almost everywhere.Consequently so is √ det DG, and hence A( Consequently for an orthonormal frame of vector fields X 1 , . . ., X n the function where is the Hodge star, because C 0 ∩ W 1,n−1 is an algebra [14,Theorem 2.1].But this is the Gauss map.

Some general facts about Busemann functions
Let us start by an elementary fact.
Lemma 23 Let S be a bounded star-shaped domain of R n with 0 as a star and with a C 1 boundary.Hence there is a parametrization of S given by where ϕ is a real valued function on S n−1 euc .Then This will be very useful for computing the Euclidean area of a Finslerian ball (hence evaluationg the Busemann function).
For the rest of this paragraph, we now suppose that there exists an R such that ∂ is a graph over B euc ( p, R) ∩ T p ∂ of height at most λ 0 for every p ∈ ∂ .This follows from C 1 regularity of the boundary (and hence uniform continuity of the derivative).Let ϕ p : T p ∂ → ∂ be this function.We denote by ν( p) the inner normal to the boundary at p. We also consider h p = ϕ p − p, ν(p) the height of ϕ( p) in the direction of ν( p).
Lemma 24 For every 0 < α ≤ 1 there is λ 0 which is independent of p, such that the cone Proof By C 1 regularity of the boundary, for every point p ∈ ∂ and every 0 < α < 1 there is a λ p < 1 such that the cone C α, p,λ p is contained in .Let B α, p,λ p be the base of this cone: This is a compact set, which is continuously parametrised by p. Consider a continuous parametrisation q) = min{|q − (q, x)| : q ∈ ∂ , x ∈ B euc (0, 1)}, which is continuous.Thus ˜ ( p) > 0 and there is a neighbourhood U p of p where ˜ is greater than 0, i.e. the distance to the base of this cone to the boundary is strictly positive for all p ∈ U p (Fig. 4).
As such for every p there is a (relatively) open subset U p ⊂ ∂ such that for every p ∈ U p , B α, p ,λ p is contained in (note that λ p is still fixed to p but p can vary).But in this case by the convexity of the corresponding cone C α, p ,λ p is contained in .Finally so is the cone C α, p ,λ for any λ p ≤ λ < 1.
In the following we will take α to be close enough to 0 such that for every p ∈ ∂ we can eventually end up in the cone C α, p,λ when travelling from the origin to p in such a cone, i.e. − p, ν(p) ≥ α| p| Lemma 25 Let λ 0 be close enough to 1 such that the cone C α, p,λ is contained in p,λ 0 for some α ≤ p/| p|, ν(p) /2, and every p.Then there is a number C which depends on the C 1 norm of ∂ such that for every p and every λ 0 ≤ λ < 1, Proof Let q 1 = λ p and let q 2 be in the normal direction to T p ∂ : Let c be a positive number such that for every p ∈ ∂ , B euc (q 1 , (1−λ)c) ⊂ C α, p,λ ∩ p,λ and B euc (q 2 , (1−λ)c) ⊂ C α, p ∩ p,λ , and hence both balls are contained in .The existence of such a c is given by the fact that we can fit a cone inside p,λ 0 .Now for any q ∈ ∂ ∩ p,λ , And so But because we can fit balls of radius c(1−λ) around both q i , it follows that |q −q i | > c(1−λ) and because uniformly over λ > λ 0 and p ∈ ∂ .And by a similar argument we have the bound Now consider the outward half spheres S + euc,ν( p) (q i ) ⊂ T q i (i = 1, 2) defined by To compare the Busemann density at the points q 1 and q 2 , we introduce the change of variables S + euc,ν( p) (q 1 ) → S + euc,ν( p) (q 2 ) obtained by first mapping v to its projection (through q 1 ) on the boundary of and then project it onto the Euclidean unit sphere centered at q 2 .We now show that this change of variables is uniformly over λ ≥ λ 0 and p ∈ ∂ bi-Lipschitz.
Let v 1 and v 2 be two elements in T q 1 .Let γ : [0, θ] → S + euc,ν( p) (q 1 ) be the great arc connecting v 1 and v 2 .Let π 1 be the projection from the unit sphere centered at q 1 , to ∂ and let π 2 that from q 2 .Consequently, if η = π 1 • γ , we have is the norm of η⊥ (q 1 − η) ( η⊥ denotes the orthogonal projection on the hyperplane perpendicular to η).The vector q 1 − η⊥ (q 1 − η) is in the tangent hyperplane at η, hence η⊥ (q 1 − η) has a norm greater than c(1 − λ) (the size of the ball we can fit around q 1 ).Now the key point is that |q 1 − q 2 | ≤ C|1 − λ|.We can apply a similar argument to bound d dt π −1 2 • η, and We denote by C, the number C 2 + C 2 c .Hence But by a similar argument Finally we open up σ (q 1 ):

Centro-projective area for the Sobolev class
The goal of this section is to prove the second main theorem: Theorem 27 Let be an open convex relatively compact subset of R n such that the boundary has a regularity W 2, p ∩ C 1 for some p ≥ n − 1.Then the Finsler volume growth entropy of is maximal, equal to n − 1 Let us first remark that we have the Sobolev embedding theorem stating that the boundary has also regularity C 1 , so that Lemma 25 can be applied in this situation.
This statement echoes the main result of [3] for which the regularity assumption is C 1,1 (hence W 2,∞ ).In fact the proof largely follows their proof: there are two crucial steps in the proof of the main theorem of [3] that need to be worked out.Precisely we want to show that (1) As a point p ∈ approaches the boundary, the Busemann density (suitably renormalized) converges almost everywhere to 2a . This is [3, Proposition 2.8] for which we recall the proof in Lemma 28.(2) We then want to use this convergence to replace the volume of balls by the centroprojective area (i.e use the Lebesgue Dominated Convergence theorem).This requires a bound of the Busemann density by a dominating L 1 function.This will use the technology of maximal functions and will be achieved in Lemma 30.
Both of the Lemmas 28 and 30 will follow from general considerations on the Busemann function (see Paragraph 4.3).
Proof We refer to Proposition 2.8] for the original proof.We use Alexandrov's theorem on the almost everywhere second differentiability of convex functions [1] [5], to yield for almost every y ∈ B euc (x, R) as t → 0, and h y : T y ∂ → R is the height of function so that {y +z +ν(y)h(z) | z ∈ T y ∂ } is a subset of ∂ (n.b. h y (0) = 0).The bilinear form D 2 h y (0) is positive semidefinite, and given almost everywhere by the weak derivative [12].It has principal values r 1 , . . .r n−1 which are the principal curvatures, with principal directions τ 1 , . . ., τ n−1 which form an orthonormal basis.For ε ∈ R there is a t 0 such that for every t < t 0 .
This implies that the parabolas Finally we can proceed as in the proof of Proposition 2.8 in [3].
We now consider a map h x : B euc (x, R) ⊂ T x ∂ → ∂ .Suppose that the restriction of h x to the line t → y + tθ is W 2, p (R).Then Lemma 29 Let {θ i |∈ N} be a countable dense set of directions in S n−2 ⊂ T x ∂ .There is a set E ⊂ B euc (x, R) of full measure such that for every y ∈ E the map f y,i : t → h(y + tθ i ) is in W 2, p (R), f y,i is twice differentiable at t = 0 and Proof This follows from the absolutely continuous on lines characterisation of Sobolev functions cf.[12].For every direction we have a set E i ⊂ B(x, R) ∩ θ ⊥ i such that the map t → h(y + tθ i is in W 2, p (R) by Fubini, and for almost every t the second derivative is given by the weak second derivative of h in the direction θ i .Let E i be the set For n = 2 and p = 1 this is bounded L We introduce several functions and g = max{1, max i g i ν(x), ν(y) } all of which are in L p (B euc (x, R/2)).Applying this inequality and Lemma 25 yields the result.
But if g ∈ L 1,∞ then √ g is integrable (on a bounded set).
Remark 31 It is worth commenting that although the centro-projective area would appear to require that is W 2, p for p ≥ (n − 1)/2, we can only get an integrable bound in L 1 (∂ ) for p ≥ n − 1.This is to be expected as this is the natural exponent for a boundary of dimension n − 1.But still the question arises whether this can be reduced.It is possible that there is a higher integrability result for convex boundaries i.e. p ≥ n − 1 − ε implies p ≥ n − 1.

Proof of Theorem 27
We can apply Lemmata 28 and 30 along with the dominated convergence theorem.This allows us to follow Eq.(26) in the proof of Theorem 3.1 in [3], and bring the limit into the integral, yielding the centro-projective area.
Acknowledgements The authors would like to thank Constantin Vernicos, and Marc Troyanov for helpful discussions, as well as anonymous referees for greatly helping us improve the exposition of this paper.
Funding Open Access funding enabled and organized by Projekt DEAL.The Funding was provided by ćole Polytechnique Fédérale de Lausanne.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Both author would like to thank the École Polytechnique Fédérale de Lausanne for hospitality and support.Second author was partially supported SNSF Grant Number 200021 163069.
max{|u|, |w|, |v| − |u| − |w|} ≤ K min{|u|, |w|, |v| − |u| − |w|}.This requires judicious application of the Ahlfors regularity, and a Vitali covering argument.First we will prove that every closed interval v contains a relative d-ball of radius |v|/3.We denote by I the unique open interval among the family I i contained in v.We divide the argument into two cases.First suppose the open interval I contained in v is bigger than |v|/3 (I is the interval leading to the 2 descendants u and w of v).Since I is smaller or equal in size to any other open interval previously appearing in the I i 's (we already obtained v by removing out bigger intervals), the open intervals on either side of v are bigger in length than |v|/3.Now v = u ∪ I ∪ w.Without loss of generality assume that |u| ≥ |w|.Then let u = [a, b] so that B d (b, |v|/3) ∩ E ⊂ v ∩ E. For the second case, suppose |I | ≤ |v|/3.Without loss of generality we may assume |u| ≥ |v|/3, and so

.
recall that κ denotes an Ahlfors s-regular measure on the Cantor set).We just have to use the fact that v contains a ball of radius |v|/3 and is contained in a ball of radius |v|.Consider some closed interval u = [a, b] with I l on its left and I r on its right.Without loss of generality assume that |I r | ≤ |I l |.We have two cases, either |I r | ≤ |u|, or |I r | > |u|.If we have the latter then B d (b, |I r |) ∩ E = u ∩ E, and 0 = κ(B d (b, |I r |)) − κ(u) ≥ c|I r | s − C|u| s , The inequality still holds if |I r | ≤ |u|.Next consider some interval v = u ∩ I ∩ w.Then κ(v) = κ(u) + κ(w) and where C is the bi-Lipschitz constant of the map.But there is a number K which depends only ont such that | f (v)| ≤ K |I |.Now given two vertices u and u in T E .Let J denote the maximal open set separating them.Then |J | ≤ d T (u , u ) ≤ |v| ≤ K |J |.Now we know that there is an x ∈ u and y ∈ u such that d(x, y) ≥ |J |, and hence C| = j log((b j − a j )e 2R/(α+1) + 1).But |b j − a j | ≤ C|ϕ(b j ) − ϕ(a j )|.Consequently we can estimate log(e 2R/(α+1) |b j − a j | + 1) ≤ log(e 2R/(α+1) C|ϕ(b j ) − ϕ(a j )| + 1)
R) | y ∈ E i and s → h(y + sθ i )is twice differentiable at t with weak derivative θ t i D 2 h(y + tθ i )θ i }.Finally E = i∈N E i .Lemma 30 Assuming is a of class C 1 ∩ W 2, p for p ≥ (n − 1), p > 1 then there is a function f ∈ L 1 (∂ ) such that (1 − λ) (n+1)/2 σ (λ p) ≤ f ( p) for all λ ≤ 1 − λ 0 .Proof This proof is inspired by[15, chapter 2, p.40].Let θ 1 , ...θ n−1 be an orthonormal basis for T x .Define the set E i to be the set of y such thath x | y+tθ i is in W 2, p (]a, b[).Again E i is of full measure.Let E = i E i .For every i define the directional maximal function.This is bounded from L p (B euc (x, R)) → L p (B euc (x, R/2)) and for 1 < p < ∞, by Fubini's theorem and the boundedness of the usual maximal function on L p (R) → L p (R).