Some Comments on the mth-Order Projection Bodies

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-Gutiérrez later gave a stability result for Petty's projection inequality. In 1970, Schneider introduced the $m$th-order setting and extended the difference body to that setting. In a previous work with J. Haddad, E. Putterman, M. Roysdon, and D. Ye [Affine isoperimetric inequalities for higher-order projection and centroid bodies, Math. Ann. 393/1 (2025) 1073–1121], we 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.