d-representability as an embedding problem

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...

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Bibliographic Details
Published in:arXiv.org 2023-07
Main Author: White, Moshe
Format: Article
Language:English
Subjects:
Convexity
Equivalence
Online Access:Get full text
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Summary: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.
ISSN:2331-8422