2024/03/08 by Dylan Langharst, Langharst, Dylan, Francisco Marín Sola +3 · 2 citations
Mathematics · #Point processes and geometric inequalities #Mathematical Inequalities and Applications
paper · pdf · doi:10.48550/arxiv.2403.05712
The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric-type inequalities for convex bodies. Following a classical work by Schneider, both inequalities have been extended to the so-called mth-order setting. In this work, we establish the mth-order analogues for these inequalities in the setting of log-concave functions. Our proof of the functional Zhang's projection inequality employs properties of the asymmetric LYZ body, significantly streamlining the argument and producing a novel approach for the case m=1. Furthermore, we introduce and analyze the radial mean bodies of a log-concave function, thereby providing a functional generalization of Gardner and Zhang's radial mean bodies. These are new even in the case m=1. Our development leverages an extension of Ball bodies, which may be of independent interest.