Barycenters of points in polytope skeleta

TL;DR

This paper explores how barycenters of points in polytope faces can be manipulated by changing face dimensions and weights, revealing limitations on prescribed barycenters and weights in high-dimensional polytopes.

Contribution

It extends previous results by allowing points in faces of dimension $k$ and $k+1$, and analyzes the constraints on prescribed barycenters and weights in polytopes.

Findings

Increasing polytope dimension by $r$ with points in $(k+1)$-faces is possible.

The gap in face dimensions for prescribed barycenters cannot exceed one.

Certain weight configurations cannot be prescribed for large $k$ in fixed $n$.

Abstract

The first author showed that for a given point $p$ in an $nk$-polytope $P$ there are $n$ points in the $k$-faces of $P$, whose barycenter is $p$. We show that we can increase the dimension of $P$ by $r$, if we allow $r$ of the points to be in $(k+1)$-faces. While we can force points with a prescribed barycenter into faces of dimensions $k$ and $k+1$, we show that the gap in dimensions of these faces can never exceed one. We also investigate the weighted analogue of this question, where a convex combination with predetermined coefficients of $n$ points in $k$-faces of an $nk$-polytope is supposed to equal a given target point. While weights that are not all equal may be prescribed for certain values of $n$ and $k$, any coefficient vector that yields a point different from the barycenter cannot be prescribed for fixed $n$ and sufficiently large $k$.