Affine chord Sobolev inequalities and radial mean bodies for functions

TL;DR

This paper introduces new affine Sobolev inequalities for functions, extending geometric inequalities to functional settings, and explores their implications for radial mean bodies and concave functions.

Contribution

It develops affine chord Sobolev inequalities for functions, generalizing previous geometric inequalities and establishing new monotonicity properties for radial mean bodies of s-concave functions.

Findings

Derived affine isoperimetric inequalities for functional radial mean bodies.

Extended affine chord Sobolev inequalities to functions, strengthening Euclidean versions.

Established monotonicity properties of radial mean bodies for s-concave 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.