Sperner’s Colorings and Optimal Partitioning of the Simplex

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 ∗, …, A k ∗) where A i ∗ contains all the points whose closest vertex is e i . We also discuss possible extensions of this result to general polytopes and some open questions.