Quantitative improvements of functional inequalities under concavity properties

TL;DR

The paper develops a unifying framework to sharpen functional inequalities for log-concave and $s$-concave functions, extending Hensley’s and Barthe–Koldobsky’s results to upper bounds for $\int_0^{\infty} t^2 f(t)\,dt$ and to general functionals $\int N(t) f(t)\,d\mu$ with convex, even $N$ and suitable measures $\mu$. A core reduction scheme confines extremals to explicit, finite-parameter families, enabling precise upper and lower bounds and equality cases, including $f(t)=c e^{-\lambda t}$ and $f(t)=c(1-\lambda t)_{+}^{1/s}$. The results yield quantitative refinements of $p$-norms and entropy for decreasing log-concave functions, probabilistic interpretations for log-concave random variables, and geometric consequences for slabs of symmetric convex bodies, isotropic constants, and floating bodies. The work thus unifies and strengthens prior inequalities, while suggesting broad extensions to general measures, concavity frameworks, and discrete settings, with implications for both analysis and convex geometry.

Abstract

A classical result of Hensley provides a sharp lower bound for the functional $\int_\mathbb{R} t^2f$, where $f$ is a non-negative, even log-concave function. In the context of studying the minimal slabs of the unit cube, Barthe and Koldobsky established a quantitative improvement of Hensley's bound. In this work, we complement their result in several directions. First, we prove the corresponding upper bound inequality for $s$-concave functions with $s\geq 0$. Second, we present a generalization of Barthe and Koldobsky's result for functionals of the form $\int_\mathbb{R} Nf\,\mathrm{d}μ$, where $N$ is a convex, even function and $μ$ belongs to a suitable class of positive Borel measures. As a consequence of the employed methods, we obtain quantitative refinements of classical inequalities for $p$-norms and for the entropy of log-concave functions. Finally, we discuss both geometric consequences and probabilistic interpretations of our results.

Figures (1)

• Figure 1: The functions $f^s$ and $\tilde{g}_{p,d}^s$ in Lemma \ref{['lemma reduction upper s']}.

Theorems & Definitions (39)

• Theorem 1: Barthe--Koldobsky
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Corollary 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof