Approximability of convex bodies and volume entropy in Hilbert geometry

Vernicos, Constantin

doi:10.2140/pjm.2017.287.223

Cited by 9 publications

(8 citation statements)

References 32 publications

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“…This was later refined by Berck, Bernig and Vernicos in [3], where they also prove the conjecture in dimension 2. Vernicos then recently proved the conjecture in dimension 3 [23].…”

Section: Volume Entropy Of Convex Domainsmentioning

confidence: 95%

Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into PSL(3,R)

Tholozan¹

2017

Duke Math. J.

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Abstract. This article studies the geometry of proper convex domains of the projective space RP n . These convex domains carry several projective invariant distances, among which the Hilbert distance d H and the Blaschke distance d B . We prove a thin inequality between those distances: for any two points x and y in a proper convex domain,We then give two interesting consequences. The first one is a conjecture of Colbois and Verovic on the volume entropy of Hilbert geometries: for any proper convex domain of RP n , the volume of a ball of radius R grows at most like e (n−1)R . The second consequence is the following fact: for any Hitchin representation ρ of a surface group into PSL(3, R), there exists a Fuchsian representation j in PSL(2, R) such that the length spectrum of j is uniformly smaller than the length spectrum of ρ. This answers positively (for n = 3) to a conjecture of Lee and Zhang.

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“…This was later refined by Berck, Bernig and Vernicos in [3], where they also prove the conjecture in dimension 2. Vernicos then recently proved the conjecture in dimension 3 [23].…”

Section: Volume Entropy Of Convex Domainsmentioning

confidence: 95%

Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into PSL(3,R)

Tholozan¹

2017

Duke Math. J.

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“…Macbeath regions enjoy many useful properties. They can be computed efficiently, they have nice packing and covering properties, and up to constant scaling factors, M (x) approximates the minimum volume cap centered at x as well as the unit balls centered at x in both the Hilbert and Blaschke geometries induced by K [11,29,30]. Macbeath regions have been introduced to computational geometry as a tool to prove lower bounds for range searching [10,14].…”

Section: Overview Of Methodsmentioning

confidence: 99%

Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

Arya¹,

Arya²,

Fonseca³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Convex bodies play a fundamental role in geometric computation, and approximating such bodies is often a key ingredient in the design of efficient algorithms. We consider the question of how to succinctly approximate a multidimensional convex body by a polytope. We are given a convex body K of unit diameter in Euclidean d-dimensional space (where d is a constant) along with an error parameter ε > 0. The objective is to determine a polytope of low combinatorial complexity whose Hausdorff distance from K is at most ε. By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. In the mid-1970's, a result by Dudley showed that O(1/ε (d−1)/2 ) facets suffice, and Bronshteyn and Ivanov presented a similar bound on the number of vertices. While both results match known worst-case lower bounds, obtaining a similar upper bound on the total combinatorial complexity has been open for over 40 years. Recently, we made a first step forward towards this objective, obtaining a suboptimal bound. In this paper, we settle this problem with an asymptotically optimal bound of O(1/ε (d−1)/2 ).Our result is based on a new relationship between ε-width caps of a convex body and its polar. Using this relationship, we are able to obtain a volume-sensitive bound on the number of approximating caps that are "essentially different." We achieve our result by combining this with a variant of the witness-collector method and a novel variable-width layered construction.

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“…has no absolutely continuous part) and try to relate the Hausdorff dimension of the support of dω to the volume entropy of (Ω, h). The paper [56] might provide some useful hints. See also [18] for a related discussion in dimension 2.…”

Section: Remarkmentioning

confidence: 99%

On Alexandrov's Surfaces with Bounded Integral Curvature

Troyanov¹

2022

Preprint

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During the years 1940-1970, Alexandrov and the "Leningrad School" have investigated the geometry of singular surfaces in depth. The theory developed by this school is about topological surfaces with an intrinsic metric for which we can define a notion of curvature, which is a Radon measure. This class of surfaces has good convergence properties and is remarkably stable with respect to various geometrical constructions (gluing etc.). It includes polyhedral surfaces as well as Riemannian surfaces of class C 2 , and both of these classes are dense families of Alexandrov's surfaces. Any singular surface that can be reasonably thought of is an Alexandrov surface and a number of geometric properties of smooth surfaces extend and generalize to this class.The goal of this paper is to give an introduction to Alexandrov's theory, to provide some examples and state some of the fundamental facts of the theory. We discuss the conformal viewpoint introduced by Yuri G. Reshetnyak and explain how it leads to a classification of compact Alexandrov's surfaces. This article is an updated and translated version of the paper [54], to be included in the forthcoming book Reshetnyak's Theory of Subharmonic Metrics edited by François Fillastre and Dmitriy Slutskiy and to be published by Springer and the Centre de recherches mathématiques (CRM) in Montréal.

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Approximability of convex bodies and volume entropy in Hilbert geometry

Cited by 9 publications

References 32 publications

Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into PSL(3,R)

Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into PSL(3,R)

Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

On Alexandrov's Surfaces with Bounded Integral Curvature

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