Isoperimetric problems for zonotopes

TL;DR

This paper generalizes Shephard's decomposition theorem for zonotopes to intrinsic volumes, solving isoperimetric problems for certain zonotopes and applying results to polarization and inequality conjectures.

Contribution

It extends the decomposition theorem to intrinsic volumes and solves specific isoperimetric problems for zonotopes generated by a small number of segments.

Findings

Solved isoperimetric problems for d- and (d+1)-segment zonotopes

Provided asymptotic estimates for large segment zonotopes

Applied results to polarization and inequality conjectures

Abstract

Shephard (Canad. J. Math. 26: 302-321, 1974) proved a decomposition theorem for zonotopes yielding a simple formula for their volume. In this note we prove a generalization of this theorem yielding similar formulas for their intrinsic volumes. We use this result to investigate geometric extremum problems for zonotopes generated by a given number of segments. In particular, we solve isoperimetric problems for d-dimensional zonotopes generated by d or d+1 segments, and give asymptotic estimates for the solutions of similar problems for zonotopes generated by sufficiently many segments. In addition, we present applications of our results to the \ell_1$ polarization problem on the unit sphere and to a vector-valued Maclaurin inequality conjectured by Brazitikos and McIntyre in 2021.