Vol. 287, No. 1, 2017

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ISSN: 0030-8730
Approximability of convex bodies and volume entropy in Hilbert geometry

Constantin Vernicos

Vol. 287 (2017), No. 1, 223–256
Abstract

The approximability of a convex body is a number which measures the difficulty in approximating that convex body by polytopes. In the interior of a convex body one can define its Hilbert geometry. We prove on the one hand that the volume entropy is twice the approximability for a Hilbert geometry in dimension two or three, and on the other hand that in higher dimensions the approximability is a lower bound of the entropy. As a corollary we solve the volume entropy upper bound conjecture in dimension three and give a new proof in dimension two different from the one given in (Pacific J. Math. 245:2 (2010), 201–225). Moreover, our method allows us to prove the existence of Hilbert geometries with intermediate volume growth on the one hand, and that in general the volume entropy is not a limit on the other hand.

Keywords
volume entropy, approximability, Hilbert geometries, Finsler metric, convex bodies
Mathematical Subject Classification 2010
Primary: 53C60
Secondary: 53C24, 58B20, 53A20
Milestones
Received: 10 June 2015
Revised: 11 July 2016
Accepted: 19 July 2016
Published: 6 February 2017
Authors
Constantin Vernicos
Institut Montpelliérain Alexander Grothendieck
Université de Montpellier
Case Courrier 051
Place Eugène Bataillon
F-34095 Montpellier Cedex
France