Some comments on the mth-order Projection Bodies
Abstract
The celebrated Petty's projection inequality is a sharp upper bound for the volume of the polar projection body of a convex body. Lutwak introduced the concept of mixed projection bodies and extended Petty's projection inequality. Alonso-Guti\'{e}rrez later did a stability result for Petty's projection inequality. In 1970, Schneider introduced the th-order setting and extended the difference body to that setting. In a previous work, we, working with Haddad, Putterman, Roysdon, and Ye, established an extension of the projection body operator to this setting. In this note, we continue this study for the mixed projection body operator as well as the question of stability.
Cite
@article{arxiv.2504.18933,
title = {Some comments on the mth-order Projection Bodies},
author = {Dylan Langharst},
journal= {arXiv preprint arXiv:2504.18933},
year = {2026}
}
Comments
To appear in the Special Issue of the Journal of Convex Analysis dedicated to Professor Alexander Plakhov