Blaschke operations on log-concave functions and affine isoperimetric inequalities

TL;DR

This paper develops a new framework of Blaschke operations on log-concave functions, leading to affine isoperimetric inequalities and a deeper understanding of affine geometric properties of these functions.

Contribution

It introduces Blaschke addition and homothety for log-concave functions, establishing their properties and applications to affine inequalities and symmetrization.

Findings

Defined canonical Blaschke sum and homothety on log-concave functions.

Proved convergence of successive Blaschke symmetrizations to a radially symmetric function.

Established affine isoperimetric inequalities and concavity properties for affine surface area.

Abstract

We introduce Blaschke addition and homothety operations on log-concave functions and study their affine-geometric consequences. Our starting point is the first variation formula of Falah and Rotem (Calc. Var. and PDE, 2026), which associates to each log-concave function a pair of surface area measures. Using the additivity of these measures, we define a canonical Blaschke sum and Blaschke homothety on the class of log-concave functions, uniquely determined up to translation. We establish the basic algebraic properties of these operations, define the associated Blaschke symmetral, and show that this symmetrization preserves both total mass and the first quermassintegral. We also prove that successive Blaschke symmetrizations converge, after translations, to a radially symmetric log-concave function, which we call the mean Blaschke symmetral. We then relate the canonical theory to projection-type constructions. In particular, we show that the functional projection body arising from the first variation coincides with the projection body of the asymmetric LYZ body, and we derive corresponding intertwining properties. As applications, we prove concavity of the entropy with respect to the canonical Blaschke sum and obtain associated Kneser--S\"uss-type inequalities. We also study a functional version of affine surface area, and prove affine isoperimetric inequalities for log-concave functions. In particular, we obtain a Blaschke-concavity property for the affine surface area and show that it is maximized, under fixed first quermassintegral, by radially symmetric functions.