Barycenters of points that are constrained to a polytope skeleton

TL;DR

This paper provides a simplified proof that any point in an n-dimensional polytope can be expressed as the barycenter of n points on its d-skeleton, extending Tverberg-type results.

Contribution

It introduces a new, simplified proof of Dobbins' result using the constraint method, advancing understanding of barycenters in polytopes.

Findings

Any point in an n-dimensional polytope is the barycenter of n points in its d-skeleton.

The proof is shorter and simpler than previous proofs.

The method extends Tverberg-type results to new settings.

Abstract

We give a short and simple proof of a recent result of Dobbins that any point in an $nd$-polytope is the barycenter of $n$ points in the $d$-skeleton. This new proof builds on the constraint method that we recently introduced to prove Tverberg-type results.