Isoperimetric problems for zonotopes
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.
Cite
@article{arxiv.2206.03204,
title = {Isoperimetric problems for zonotopes},
author = {Antal Joós and Zsolt Lángi},
journal= {arXiv preprint arXiv:2206.03204},
year = {2023}
}
Comments
24 pages, 2 figures