Affine chord Sobolev inequalities and radial mean bodies for functions

Abstract

Affine isoperimetric inequalities for the functional radial mean bodies are derived from the new affine chord Sobolev inequalities, which extend the recent affine isoperimetric inequalities of Haddad and Ludwig from convex bodies to functions. The affine chord Sobolev inequalities further imply a strengthening of the Euclidean chord Sobolev inequalities introduced by Ba\^eta and Cai. Moreover, for $s$-concave functions $f$ with compact support and $s>0$, a parameter-dependent monotonicity property of the functional radial mean body $R_{\alpha}f$ is obtained: $R_\alpha f \subset $R_\beta f$ for $-1<\alpha < \beta$, and, after suitable normalization, the reverse inclusion also holds. These sharp results generalize the corresponding monotonicity for geometric radial mean bodies established by Gardner and Zhang.