Affine chord Sobolev inequalities and radial mean bodies for functions

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 Baêta 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.