Sperner's colorings and optimal partitioning of the simplex

TL;DR

This paper explores optimal coloring and partitioning strategies of the simplex related to Sperner's Lemma, focusing on minimizing boundary areas and the number of non-monochromatic cells, with implications for computational geometry.

Contribution

It establishes new lower bounds on non-monochromatic cells, constructs Sperner-admissible labelings with limited colors per cell, and identifies Voronoi partitions as area minimizers.

Findings

Lower bound on non-monochromatic cells in Sperner-admissible labelings

Existence of Sperner-admissible labelings with at most 4 colors per cell

Voronoi partition minimizes boundary surface area in Sperner-admissible partitions

Abstract

We discuss coloring and partitioning questions related to Sperner's Lemma, originally motivated by an application in hardness of approximation. Informally, we call a partitioning of the $(k-1)$-dimensional simplex into $k$ parts, or a labeling of a lattice inside the simplex by $k$ colors, "Sperner-admissible" if color $i$ avoids the face opposite to vertex $i$. The questions we study are of the following flavor: What is the Sperner-admissible labeling/partitioning that makes the total area of the boundary between different colors/parts as small as possible? First, for a natural arrangement of "cells" in the simplex, we prove an optimal lower bound on the number of cells that must be non-monochromatic in any Sperner-admissible labeling. This lower bound is matched by a simple labeling where each vertex receives the minimum admissible color. Second, we show for this arrangement that in contrast to Sperner's Lemma, there is a Sperner-admissible labeling such that every cell contains at most $4$ colors. Finally, we prove a geometric variant of the first result: For any Sperner-admissible partition of the regular simplex, the total surface area of the boundary shared by at least two different parts is minimized by the Voronoi partition $(A^*_1,\ldots,A^*_k)$ where $A^*_i$ contains all the points whose closest vertex is $i$. We also discuss possible extensions of this result to general polytopes and some open questions.