d-representability as an embedding problem

TL;DR

This paper establishes an equivalence between d-representability of simplicial complexes and the existence of certain maps into R^d, providing a new topological framework for analyzing intersection patterns of convex sets.

Contribution

It introduces a novel equivalence linking d-representability to map existence, enabling topological methods to study intersection patterns of convex sets.

Findings

Equivalence between d-representability and specific map existence.

Framework for using topological tools like Borsuk-Ulam theorem.

Potential to prove or disprove d-representability using topology.

Abstract

An abstract simplicial complex is said to be $d$-representable if it records the intersection pattern of a collection of convex sets in $\mathbb{R}^d$. In this paper, we show that $d$-representability of a simplicial complex is equivalent to the existence of a map with certain properties, from a closely related simplicial complex into $\mathbb{R}^d$. This equivalence suggests a framework for proving (and disproving) $d$-representability of simplicial complexes using topological methods such as applications of the Borsuk-Ulam theorem, which we begin to explore.