Functional John and L\"owner conditions for pairs of log-concave functions

TL;DR

This paper extends classical geometric optimization problems to the setting of log-concave functions, providing new characterizations for the largest and smallest integral positions under pointwise constraints, thus generalizing John and L"owner conditions.

Contribution

It introduces a unified framework for functional John and L"owner problems, extending known geometric results to a broader class of log-concave functions and exploring their dual relationships.

Findings

Characterization of solutions for the functional John problem.

Characterization of solutions for the functional L"owner problem.

Analysis of the relationship between John and L"owner problems in the functional setting.

Abstract

John's fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ in $\mathbb{R}^d$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by positions (affine images) of another body $L$. Another, more recent direction is to consider logarithmically concave functions on $\mathbb{R}^d$ instead of convex bodies: we designate some special, radially symmetric log-concave function $g$ as the analogue of the Euclidean ball, and want to find its largest integral position under the constraint that it is pointwise below some given log-concave function $f$. We follow both directions simultaneously: we consider the functional question, and allow essentially any meaningful function to play the role of $g$ above. Our general theorems jointly extend known results in both directions. The dual problem in the setting of convex bodies asks for the smallest volume ellipsoid, called \emph{L{\"o}wner's ellipsoid}, containing $K$. We consider the analogous problem for functions: we characterize the solutions of the optimization problem of finding a smallest integral position of some log-concave function $g$ under the constraint that it is pointwise above $f$. It turns out that in the functional setting, the relationship between the John and the L{\"o}wner problems is more intricate than it is in the setting of convex bodies.