article

Title: Minimizing the mass of the codimension two skeleton of convex, unit volume polyhedra

Authors: Ryan C. Scott

Issue: Volume 61 (2012), Issue 4, 1513-1564

Abstract:

$In this paper we establish the existence and partial regularity of a $(d-2)$-dimensional edge length minimizing polyhedron in $\mathbb{R}^d$. The minimizer is a generalized convex polytope of volume $1$ which is the limit of a minimizing sequence of polytopes converging in the Hausdorff metric. We show that the $(d-2)$-dimensional edge length is lower-semicontinuous under this sequential convergence. Here the edge set of the limit generalized polytope is a closed subset of the boundary whose complement in the boundary consists of countably many relatively open planar regions. The case $d=3$ was previously obtained by S. Berger [S. Berger, \textit{Edge length minimizing polyhedra}, Ph.D. thesis (2002), Rice University].$