Approximation Depth of Convex Polytopes

TL;DR

This paper investigates how well convex polytopes, especially simplices, can be approximated using Minkowski sums and unions, revealing limitations on approximating simplices and characterizing simplices as unique outer additive bodies.

Contribution

It introduces a framework for approximating polytopes via Minkowski sums and unions, and characterizes simplices as the only outer additive convex bodies, highlighting their approximation limitations.

Findings

Simplices can only be trivially approximated.

Characterization of simplices as the only outer additive convex bodies.

Limitations on approximating polytopes of a given depth.

Abstract

We study approximations of polytopes in the standard model for computing polytopes using Minkowski sums and (convex hulls of) unions. Specifically, we study the ability to approximate a target polytope by polytopes of a given depth. Our main results imply that simplices can only be ``trivially approximated''. On the way, we obtain a characterization of simplices as the only ``outer additive'' convex bodies.