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 later introduced the concept of mixed projection bodies, and extended Petty's projection inequality. Alonso-Guti\'{e}rrez later did a type of stability 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, we, working with Haddad, Putterman, Roysdon and Ye, established an extension of the projection operator to this setting. In this note, we continue this study for the mixed projection body operator as well as the question of stability.