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Extremal affine surface areas in a functional setting

Stephanie Egler, Elisabeth M. Werner

Abstract

We introduce extremal affine surface areas in a functional setting. We show their main properties. Among them are linear invariance, isoperimetric inequalities and monotonicity properties. We establish a new duality formula, which shows that the maximal (resp. minimal) inner affine surface area of an $s$-concave function on $\mathbb{R}^n$ equals the maximal (resp. minimal) outer affine surface area of its Legendre polar. We estimate the ``size" of these quantities: up to a constant depending on $n$ and $s$ only, the extremal affine surface areas are proportional to a power of the integral of $f$. This extends results obtained in the setting of convex bodies. We recover and improve those as a corollary to our results.

Extremal affine surface areas in a functional setting

Abstract

We introduce extremal affine surface areas in a functional setting. We show their main properties. Among them are linear invariance, isoperimetric inequalities and monotonicity properties. We establish a new duality formula, which shows that the maximal (resp. minimal) inner affine surface area of an -concave function on equals the maximal (resp. minimal) outer affine surface area of its Legendre polar. We estimate the ``size" of these quantities: up to a constant depending on and only, the extremal affine surface areas are proportional to a power of the integral of . This extends results obtained in the setting of convex bodies. We recover and improve those as a corollary to our results.
Paper Structure (24 sections, 13 theorems, 186 equations)

This paper contains 24 sections, 13 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Notation.
  3. Background
  4. $s$-concave functions.
  5. An associated body.
  6. $\lambda$-affine surface area for $s$-concave functions.
  7. Extremal affine surface areas for functions
  8. Definition.
  9. Main results
  10. Basic properties of the extremal surface areas.
  11. The "size" of the extremal affine surface areas.
  12. The cases: $\frac{1}{s} \in \mathbb{N}$ and $0 < s \leq 1$.
  13. The case $s \geq 1$.
  14. Results for convex bodies.
  15. Proofs
  16. ...and 9 more sections

Key Result

Proposition 2

CFGLSW Let $s > 0$ and $f \in C_s(\mathbb{R}^n)$. (i) For all linear maps $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ and for all $\lambda \in \mathbb{R}$, Moreover, for all $\alpha \in \mathbb{R}$, $\alpha > 0$, $as_\lambda^{(s)}( \alpha \, f ) = \alpha^{1-2 \lambda}\, as_\lambda ^{(s)}(f)$. (ii) Let $f \in {\mathit C}_s^+(\mathbb R^n)$. Then Equality holds in the first inequality if $\lambda=0

Theorems & Definitions (25)

  • Definition 1
  • Proposition 2
  • Lemma 3
  • Definition 4
  • Proposition 5
  • proof
  • Theorem 6
  • Lemma 7
  • proof
  • Theorem 8
  • ...and 15 more