The typical irregularity of virtual convex bodies

TL;DR

This paper investigates the structure of virtual convex bodies, revealing that in the typical case, these objects are highly irregular and singular, especially when considering strictly convex bodies in higher dimensions.

Contribution

It provides a geometric interpretation of virtual convex bodies for strictly convex bodies and demonstrates their typical irregularity in a Baire category sense.

Findings

Virtual convex bodies are highly singular in the typical case.

Geometric interpretation involves differences of boundary points with the same outer normal.

Results extend understanding of convex bodies beyond polytopes and planar cases.

Abstract

The semigroup of convex bodies in ${\mathbb R}^n$ with Minkowski addition has a canonical embedding into an abelian group; its elements have been called virtual convex bodies. Geometric interpretations of such virtual convex bodies have been particularly fruitful under the restriction to polytopes. For general convex bodies, mainly the planar case has been studied, as a part of the more general investigation of hedgehogs. Here we restrict ourselves to strictly convex bodies in ${\mathbb R}^n$. A particularly natural geometric interpretation of virtual convex bodies can then be seen in the set of differences of boundary points of two convex bodies with the same outer normal vector. We describe how in the typical case (in the sense of Baire category) this leads to a highly singular object.