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On maximal intersection position for logarithmically concave functions and measures

Steven Hoehner, Michael Roysdon

TL;DR

The paper extends the notion of maximal intersection position from convex bodies to pairs of log-concave functions and even log-concave measures, formulating a functional analogue via $P_{f,g}(T,z)$ and proving a John-type decomposition of the identity. A central contribution is establishing the existence of optimizers for the maximal intersection problem, then deriving variational formulas that yield isotropic-type conditions and a two-term decomposition separating nonsingular and boundary contributions; these results hold under weak hypotheses and even permit unbounded supports. The authors further apply the framework to define a John-type position for even log-concave measures, deriving isotropic boundary measures and linking to classical John theory in a functional setting. Together, these results generalize John-type decompositions to the functional/measure context, enabling a principled analysis of log-concave objects in high dimensions and recovering special cases corresponding to indicators of convex bodies.

Abstract

A new position is introduced and studied for the convolution of log-concave functions, which may be regarded as a functional analogue of the maximum intersection position of convex bodies introduced and studied by Artstein-Avidan and Katzin (2018) and Artstein-Avidan and Putterman (2022). Our main result is a John-type theorem for the maximal intersection position of a pair of log-concave functions, including the corresponding decomposition of the identity. The main result holds under very weak assumptions on the functions; in particular, the functions considered may both have unbounded supports. As an application of our results, we introduce a John-type position for even $\log$-concave measures.

On maximal intersection position for logarithmically concave functions and measures

TL;DR

The paper extends the notion of maximal intersection position from convex bodies to pairs of log-concave functions and even log-concave measures, formulating a functional analogue via and proving a John-type decomposition of the identity. A central contribution is establishing the existence of optimizers for the maximal intersection problem, then deriving variational formulas that yield isotropic-type conditions and a two-term decomposition separating nonsingular and boundary contributions; these results hold under weak hypotheses and even permit unbounded supports. The authors further apply the framework to define a John-type position for even log-concave measures, deriving isotropic boundary measures and linking to classical John theory in a functional setting. Together, these results generalize John-type decompositions to the functional/measure context, enabling a principled analysis of log-concave objects in high dimensions and recovering special cases corresponding to indicators of convex bodies.

Abstract

A new position is introduced and studied for the convolution of log-concave functions, which may be regarded as a functional analogue of the maximum intersection position of convex bodies introduced and studied by Artstein-Avidan and Katzin (2018) and Artstein-Avidan and Putterman (2022). Our main result is a John-type theorem for the maximal intersection position of a pair of log-concave functions, including the corresponding decomposition of the identity. The main result holds under very weak assumptions on the functions; in particular, the functions considered may both have unbounded supports. As an application of our results, we introduce a John-type position for even -concave measures.
Paper Structure (11 sections, 18 theorems, 115 equations)

This paper contains 11 sections, 18 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Maximal intersection position for pairs of log-concave functions
  3. Existence of the maximal intersection position
  4. John-type theorem for the maximal intersection position
  5. Proof of Theorem \ref{['t:exitence']}
  6. Variational formulas and the proof of Theorem \ref{['t:isotropic']}
  7. Proof of Theorems \ref{['t:variationalformula']} and \ref{['t:shifts']}
  8. Compactly supported functions
  9. The general case
  10. Proof of Theorem \ref{['t:shifts']}
  11. An application to a John-type position for even log-concave measures

Key Result

Theorem A

Let $K$ be a convex body in $\mathbb{R}^n$ such that $B_2^n\subset K$. Then the following are equivalent:

Theorems & Definitions (38)

  • Theorem A: John's Theorem JohnBall92Ball97
  • Theorem B
  • Definition 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Definition 8
  • ...and 28 more