  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.
