Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions

The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric inequalities relating the volumes of a convex body and its difference body and polar projection body, respectively. Following a classical work by Schneider, both inequalities have been extended to the so-called higher-order setting. In this work, we establish the higher-order analogues for these inequalities in the setting of log-concave functions. In particular, this extends the Zhang's inequality for absolutely continuous log-concave functions. We introduce an iterated sup-convolution to tackle the Rogers-Shephard inequality.