Chord Sobolev inequalities

TL;DR

This work develops the chord Sobolev inequalities, a new analytic family for $\alpha>-1$ that augment the classical and fractional Sobolev frameworks and connect to chord isoperimetric inequalities in integral geometry. It unifies geometric chord power integrals with functional analogues through level-set decompositions, Minkowski-type inequalities, and dual mixed volumes, and introduces the radial mean body machinery for functions. The paper identifies endpoint limits at $\alpha\to0^+$ and $\alpha\to n$, yielding chord entropy inequalities and a logarithmic Sobolev-type inequality, thereby completing the theory in the spirit of BBM and Almgren–Lieb. The results bridge convex geometry, integral geometry, and analysis of log-concave densities, with applications to entropy of random lines and new functional inequalities for broad function classes. Overall, the work provides a comprehensive geometric–analytic framework that extends Sobolev-type inequalities to a broader class of chord-analytic objects and their limiting entropy regimes.

Abstract

The paper establishes a new family of 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. The paper also demonstrates the connection of all these inequalities to the chord isoperimetric inequalities in integral geometry. The limiting cases of the chord Sobolev inequalities are derived as well, one of which can be viewed as a logarithmic Sobolev-type inequality. These limiting cases, combined with the work of Bourgain, Brezis and Mironescu, complete the theory of chord Sobolev inequalities.

Theorems & Definitions (51)

• Theorem 2.1: Riesz's rearrangement inequality.
• Theorem 2.2: Burchard
• Lemma 3.1
• proof
• proof : Proof of Theorem A
• Lemma 3.2
• proof
• proof : Proof of Theorem B
• Lemma 4.1
• proof