Thrifty approximations of convex bodies by polytopes

TL;DR

This paper presents nearly optimal methods for approximating convex bodies in high-dimensional spaces with polytopes that have minimal vertices, depending on parameters like epsilon, tau, and the body's symmetry.

Contribution

It introduces constructions for approximating convex bodies with polytopes that are close to optimal in vertex count for various geometric conditions.

Findings

Approximation with roughly epsilon^{-d/2} vertices for symmetric bodies and tau=1+epsilon.

Approximation with roughly d^{1/epsilon} vertices for tau=sqrt{epsilon d}.

General bounds for bodies with -C in mu C, depending on mu and tau.

Abstract

Given a convex body C in R^d containing the origin in its interior and a real number tau > 1 we seek to construct a polytope P in C with as few vertices as possible such that C in tau P. Our construction is nearly optimal for a wide range of d and tau. In particular, we prove that if C=-C then for any 1>epsilon>0 and tau=1+epsilon one can choose P having roughly epsilon^{-d/2} vertices and for tau=sqrt{epsilon d} one can choose P having roughly d^{1/epsilon} vertices. Similarly, we prove that if C in R^d is a convex body such that -C in mu C for some mu > 1 then one can choose P having roughly ((mu+1)/(tau-1))^{d/2} vertices provided (tau-1)/(mu+1) << 1.