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An extremal property of the symmetric decreasing rearrangement

Steven Hoehner, Júlia Novaes

Abstract

It is shown that for a given log-concave function, its symmetric decreasing rearrangement is always harder to approximate in the symmetric difference metric by inner log-linearizations with a fixed number of break points. This extends a classical result of Macbeath (1951) from convex bodies to a functional setting.

An extremal property of the symmetric decreasing rearrangement

Abstract

It is shown that for a given log-concave function, its symmetric decreasing rearrangement is always harder to approximate in the symmetric difference metric by inner log-linearizations with a fixed number of break points. This extends a classical result of Macbeath (1951) from convex bodies to a functional setting.
Paper Structure (13 sections, 7 theorems, 71 equations)

This paper contains 13 sections, 7 theorems, 71 equations.

Table of Contents

  1. Introduction and main results
  2. Overview of the paper
  3. Background and notation
  4. Log-concave functions
  5. Inner linearizations and piecewise affine approximation of convex functions
  6. Symmetric decreasing rearrangements
  7. Steiner symmetrizations
  8. Proof of Theorem \ref{['mainThm']}
  9. Extension to $\alpha$-concave functions
  10. Appendix: further applications of Theorem \ref{['general-thm']}
  11. Isoperimetric inequality for total variation.
  12. Second moment (moment of inertia).
  13. Hardy--Littlewood-type inequality with radial weight.

Key Result

Theorem 1.1

Fix integers $n\geq 1$ and $N\geq n+2$, and consider the functional $G_{n,N}:\mathop{\mathrm{LC}}\nolimits_{\rm c}(\mathbb{R}^n)\to [0,\infty)$ defined by Then for every $f\in\mathop{\mathrm{LC}}\nolimits_{\rm c}(\mathbb{R}^n)$, we have

Theorems & Definitions (16)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • ...and 6 more