Isoperimetric problems for zonotopes

Abstract

AbstractShephard (Canad. J. Math. 26 (1974), 302–321) 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 formulae 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 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 ℓ1 polarization problem on the unit sphere and to a vector‐valued Maclaurin inequality conjectured by Brazitikos and McIntyre in 2021.