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On Minkowski symmetrizations of $α$-concave functions and related applications

Steven Hoehner

TL;DR

This work extends classical geometric symmetrization to the functional setting by introducing an α-parametrized Minkowski symmetral for α-concave functions and establishing hypo-convergence to a rotation-invariant hypo-symmetrization. It proves that successive Minkowski symmetrizations yield a limiting hypo-symmetrized function and uses this framework to show that hypo-symmetrization is harder to approximate by inner log-linearizations than the original function, mirroring a mean-width extremal phenomenon for convex bodies. A broad extremal property and a functional Urysohn-type inequality follow from these constructions, situating the results within the broader “geometrization of probability” program. The results provide a robust functional-analytic toolkit for mean width, approximation by inner linearizations, and symmetry-driven extremals in the space of α-concave functions.

Abstract

A Minkowski symmetral of an $α$-concave function is introduced, and some of its fundamental properties are derived. It is shown that for a given $α$-concave function, there exists a sequence of Minkowski symmetrizations that hypo-converges to its ``hypo-symmetrization". As an application, it is shown that the hypo-symmetrization of a log-concave function $f$ is always harder to approximate than $f$ is by ``inner log-linearizations" with a fixed number of break points. This is a functional analogue of the classical geometric result which states that among all convex bodies of a given mean width, a Euclidean ball is hardest to approximate by inscribed polytopes with a fixed number of vertices. Finally, a general extremal property of the hypo-symmetrization is deduced, which includes a Urysohn-type inequality and the aforementioned approximation result as special cases.

On Minkowski symmetrizations of $α$-concave functions and related applications

TL;DR

This work extends classical geometric symmetrization to the functional setting by introducing an α-parametrized Minkowski symmetral for α-concave functions and establishing hypo-convergence to a rotation-invariant hypo-symmetrization. It proves that successive Minkowski symmetrizations yield a limiting hypo-symmetrized function and uses this framework to show that hypo-symmetrization is harder to approximate by inner log-linearizations than the original function, mirroring a mean-width extremal phenomenon for convex bodies. A broad extremal property and a functional Urysohn-type inequality follow from these constructions, situating the results within the broader “geometrization of probability” program. The results provide a robust functional-analytic toolkit for mean width, approximation by inner linearizations, and symmetry-driven extremals in the space of α-concave functions.

Abstract

A Minkowski symmetral of an -concave function is introduced, and some of its fundamental properties are derived. It is shown that for a given -concave function, there exists a sequence of Minkowski symmetrizations that hypo-converges to its ``hypo-symmetrization". As an application, it is shown that the hypo-symmetrization of a log-concave function is always harder to approximate than is by ``inner log-linearizations" with a fixed number of break points. This is a functional analogue of the classical geometric result which states that among all convex bodies of a given mean width, a Euclidean ball is hardest to approximate by inscribed polytopes with a fixed number of vertices. Finally, a general extremal property of the hypo-symmetrization is deduced, which includes a Urysohn-type inequality and the aforementioned approximation result as special cases.
Paper Structure (24 sections, 21 theorems, 115 equations)

This paper contains 24 sections, 21 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background on convex sets in $\mathbb{R}^n$
  4. Background on convex functions and log-concave functions
  5. Operations on ${\rm Conv}(\mathbb{R}^n)$ and $\mathop{\mathrm{LC}}\nolimits(\mathbb{R}^n)$
  6. Properties of the Legendre--Fenchel conjugate
  7. Quasiconcave functions and the layer cake definition of mean width
  8. Algebraic operations on $\alpha$-concave functions
  9. The $\alpha$-support function and $\alpha$-mean width of an $\alpha$-concave function
  10. Minkowski symmetrizations of functions
  11. The $\alpha$-Minkowski symmetral of an $\alpha$-concave function
  12. Properties of the $\alpha$-Minkowski symmetral
  13. The layer cake definition of a Minkowski symmetral
  14. The Minkowski symmetral of a convex function
  15. The base function of an $\alpha$-Minkowski symmetral
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $\psi_j,\psi,\varphi\in{\rm Conv}(\mathbb{R}^n)$ be proper and lower semicontinuous. Then, the following properties hold true:

Theorems & Definitions (57)

  • Lemma 2.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.3
  • Remark 2.5
  • Lemma 2.6
  • Definition 3.1
  • ...and 47 more