An intrinsic volume metric for the class of convex bodies in $\mathbb{R}^n$

TL;DR

This paper introduces a new intrinsic volume metric for convex bodies in bR^n and demonstrates its effectiveness by improving approximation bounds for the Euclidean ball with polytopes, revealing a dimension-dependent factor.

Contribution

It proposes a novel intrinsic volume metric and applies it to derive improved approximation inequalities for convex bodies in bR^n.

Findings

The new metric leads to better approximation bounds for the Euclidean ball.

Dropping containment restrictions enhances approximation estimates by a factor related to dimension.

The phenomenon observed extends previous results in volume, surface area, and mean width approximations.

Abstract

A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width approximation of the ball.