Asymmetric Blaschke-Santaló functional inequalities

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 Lp functional analogue to the center of mass that is necessary for our arguments and that might be of independent interest.

Sharp affine Trudinger–Moser inequalities: A new argument

“…To help explaining our method, in this subsection, we will recall a proof of the sharp affine Sobolev inequalities [51] using the Sobolev type inequalities with arbitrary norm and the Busemann-Petty centroid inequality. This proof could be found in [25,26]. Let : R → [0, ∞) be a convex function satisfying the homogeneity property ( ) = | | ( ), ∀ ( , ) ∈ R × R. Futhermore, we assume that | | ≤ ( ) ≤ | | , ∀ ∈ R for some positive constants ≤ .…”

Blaschke-Santaló inequalities for Minkowski and Asplund endomorphisms

“…The cases for equality in (1.8) were settled by Artstein-Avidan, Klartag and Milman [7], who also established a far-reaching extension of (1.8) to not necessarily even functions (cf. Section 2), that has sparked a great deal of research interest in recent years, see [6,9,22,27,[34][35][36]48].…”

The functional inequality for the mixed quermassintegral

“…Motivated by the analogy properties between the log-concave functions and the volume of convex bodies in K n , the interest in studying the log-concave functions has been considerably increasing. For example, the functional Blaschke-Santaló inequality for even logconcave function is discussed by Ball in [6,7]; for the general case see [8,17,21,28]. The mean width for a log-concave function is introduced by Klartag, Milman and Rotem (see [22,26,27]).…”