arXiv subjects

Xiaxing Cai

Publications and source records attributed to Xiaxing Cai.

4 recordsLinked to original sources

arXiv2026

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 $R_α f$ is obtained: $R_αf \subset R_βf$ for $-1 < α< β$, 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.

math.MG↗

arXiv2026

Affine Logarithmic HLS and Beckner-Type Logarithmic Sobolev Inequalities

In this paper, we consider two limiting cases ($α\rightarrow n$ and $α\rightarrow 0 $) of the recent affine HLS inequalities by Haddad and Ludwig. As $α\rightarrow n$, the affine logarithmic HLS inequality is established, which is stronger than the logarithmic HLS inequality by Carlen and Loss from 1992 and Beckner from 1993. As $α\rightarrow 0$, an affine version of Beckner's logarithmic Sobolev inequality is established, which is also a limiting case of the affine fractional $L^2$ Sobolev inequalities. The affine logarithmic Sobolev inequality is stronger than the original version by Beckner from 1995.

math.MG↗

arXiv2026

Chord Sobolev inequalities

The paper establishes a new family of sharp analytic inequalities. Together with the fractional Sobolev inequalities of Almgren and Lieb, they form a complete class of analytic inequalities, referred to as the chord Sobolev inequalities. A close connection between these inequalities and chord isoperimetric inequalities in integral geometry is established through a functional extension of chord power integrals. The limiting cases of the chord Sobolev inequalities are derived, one of which yields a logarithmic Sobolev-type inequality. Combined with the work of Bourgain, Brezis, and Mironescu, these results complete the picture of the chord Sobolev inequalities, including their endpoint cases.

math.MG↗

arXiv2025

Anisotropic fractional area measures

The anisotropic $s$-fractional area measures are introduced as the first variation of the anisotropic fractional $s$-perimeter $P_s(K,L)$, with $L$ an origin symmetric convex body and $s\in(0,1)$. As $s\rightarrow 1^-$, the anisotropic $s$-fractional area measure converges to the mixed area measure of $K$ and the moment body of $L$. The Minkowski problem of these measures are solved. Finally, a necessary condition for the convexity of optimizers in the anisotropic fractional isoperimetric inequality is derived.

math.MG↗