Affine Chord Sobolev Inequalities from Integral Geometry

TL;DR

This work develops functional analogues of isoperimetric inequalities for chord power integrals $I_{oldsymbol{ abla}+1}$, establishing sharp chord-Sobolev inequalities for α∈(0,n) and comprehensive affine refinements. It introduces the 1-homogeneous functional framework via $I_{oldsymbol{ abla}+1}(f)$ and the anisotropic star-shaped set ${ m L}_{oldsymbol{ abla}} f$, connecting to dual mixed volumes and radial mean bodies. The paper further extends these ideas to radial mean bodies for log-concave and $s$-concave functions, proving reversed and sharp inclusions with explicit equality cases tied to simplex-based exponentials. It synthesizes geometric and functional perspectives through fractional polar projection bodies and affine Petty-type inequalities, yielding stronger affine versions than their Euclidean counterparts and clarifying the roles of level sets and symmetrization. Overall, the results provide a unified, affine-invariant picture for chord-based inequalities in both the geometric and functional settings, with precise equality characterizations and limiting connections to classical inequalities.

Abstract

Functional analogues of isoperimetric inequalities for chord power integrals are established for $α>0$, together with a complete characterization of the equality cases. For $α\in(0,n)$, these inequalities -- referred to as chord Sobolev inequalities -- serves as a counterpart to the fractional Sobolev inequality of Almgren and Lieb, which corresponds to the functional isoperimetric inequality for chord power integrals with $α\in(-1,0)$. . Moreover, affine isoperimetric inequalities for chord power integrals are derived, that are significantly stronger than their Euclidean counterparts. Extending the framework to functional and affine settings, functional extensions of chord power integrals and radial mean bodies are introduced. The inclusion relations for classical radial mean bodies due to Gardner and Zhang are extended to both log-concave and $s$-concave function, and the extensions are shown to be sharp.

Theorems & Definitions (48)

• Theorem 2.1: Riesz's rearrangement inequality.
• Theorem 2.2: Burchard
• Lemma 2.3
• Theorem 2.4
• Lemma 3.1
• proof
• proof : Proof of Theorem 1
• Proposition 4.1
• proof
• Lemma 5.1