Asymmetric Blaschke-Santaló functional inequalities

TL;DR

This work extends the Blaschke-Santaló inequality to asymmetric, functional contexts by developing an $L_p$-type analysis within convex geometry. The authors introduce an $L_p$-functional analogue of the center of mass, establish the existence of an $L_p$-functional Santaló point $c_{f,p}$, and prove two main asymmetric inequalities—one symmetric and one translated by $c_{f,p}$—with sharp constants and explicit equality cases. The methods combine $M_{\varepsilon,p}$-type constructions, $L_p$ centroid theory, and a Brouwer degree argument, yielding results that recover the classical BS inequality and the Lutwak–Yang–Zhang inequality while enriching the affine-invariant, functional toolkit. These results deepen connections between convex geometry, functional inequalities, and information-theoretic bounds, with potential implications for probability and analysis on spaces of densities.

Abstract

In this work we establish functional asymmetric versions of the celebrated Blaschke-Santaló inequality. As consequences of these inequalities we recover their geometric counterparts with equality cases, as well as, another inequality with strong probabilistic flavour that was firstly obtained by Lutwak, Yang and Zhang. We present a brief study on an $L_p$ functional analogue to the center of mass that is necessary for our arguments and that might be of independent interest.

Theorems & Definitions (12)

• Lemma 1.1: see Sch
• Lemma 1.2: RENYI
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Theorem 2.2
• Remark 2.3
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm_existeSantaloLp']}