Approximability of convex bodies and volume entropy in Hilbert geometry
arXiv:1207.1342 · doi:10.2140/pjm.2017.287.223
Abstract
The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three and that in higher dimension it is a lower bound of the entropy. As a corollary we solve the entropy upper bound conjecture in dimension three and give a new proof in dimension two from the one found in Berck-Bernig-Vernicos (arXiv:0810.1123v2, published).
arXiv comment33 pages, 7 figures. Exposition improved, paper accepted for publication in pacific