# Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes

## TL;DR

This paper investigates the asymptotic approximation of the Euclidean ball by polytopes using intrinsic and dual volume-based distances, providing sharp bounds and identifying nearly optimal polytopes in high dimensions.

## Contribution

It introduces new notions of distance between convex bodies based on intrinsic and dual volumes and derives asymptotically sharp bounds for high-dimensional approximation.

## Key findings

- Established asymptotic bounds for approximation by polytopes using intrinsic volumes.
- Identified a polytope nearly optimal for all intrinsic volumes simultaneously.
- Derived asymptotic formulas for approximation using dual volumes from Lutwak's theory.

## Abstract

We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional, and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope which is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants. Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak's dual Brunn-Minkowski theory.