On the vertex index of convex bodies

TL;DR

This paper introduces the vertex index of centrally symmetric convex bodies, linking it to illumination parameters and covering conjectures, providing asymptotic and dimension-specific estimates.

Contribution

It defines the vertex index and establishes sharp asymptotic bounds, connecting it to illumination and covering problems in convex geometry.

Findings

Asymptotically sharp estimates of vertex index in general case

Sharp estimates for dimensions 2 and 3

Connection between vertex index and illumination parameter

Abstract

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by 2^d smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. Also, we provide sharp estimates in dimensions 2 and 3.