Minkowski complexes and convex threshold dimension

TL;DR

This paper investigates the convex threshold dimension of simplicial complexes, showing it can be arbitrarily large, and highlights the importance of convexity in Minkowski complex realizations.

Contribution

It introduces the concept of convex threshold dimension and demonstrates its unboundedness, connecting it to threshold graph theory and emphasizing convexity's role.

Findings

Convex threshold dimension can be arbitrarily large.

Every simplicial complex can be realized as a Minkowski complex.

Convexity is essential for the properties studied.

Abstract

For a collection of convex bodies $P_1,\dots,P_n \subset \mathbb{R}^d$ containing the origin, a Minkowski complex is given by those subsets whose Minkowski sum does not contain a fixed basepoint. Every simplicial complex can be realized as a Minkowski complex and for convex bodies on the real line, this recovers the class of threshold complexes. The purpose of this note is the study of the convex threshold dimension of a complex, that is, the smallest dimension in which it can be realized as a Minkowski complex. In particular, we show that the convex threshold dimension can be arbitrarily large. This is related to work of Chv\'atal and Hammer (1977) regarding forbidden subgraphs of threshold graphs. We also show that convexity is crucial this context.