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Weighted floating functions and weighted functional affine surface areas

Carsten Schütt, Christoph Thaele, Nicola Turchi, Elisabeth M. Werner

Abstract

The purpose of this paper is to introduce the new concept of weighted floating functions associated with log concave or $s$-concave functions. This leads to new notions of weighted functional affine surface areas. Their relation to more traditional versions of functional affine surface areas as well as to the classical affine surface areas for convex bodies is discussed in detail.

Weighted floating functions and weighted functional affine surface areas

Abstract

The purpose of this paper is to introduce the new concept of weighted floating functions associated with log concave or -concave functions. This leads to new notions of weighted functional affine surface areas. Their relation to more traditional versions of functional affine surface areas as well as to the classical affine surface areas for convex bodies is discussed in detail.
Paper Structure (12 sections, 13 theorems, 132 equations)

This paper contains 12 sections, 13 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Conventions on notation
  3. Weighted floating sets and functions
  4. Weighted floating sets
  5. Convex and log concave functions and their floating functions
  6. $s$-concave functions and their floating functions
  7. Main Theorems and their consequences
  8. The case of log concave functions
  9. The case of $s$-concave functions
  10. Proofs
  11. Proof of Theorems \ref{['theo:f-deltafloat2']} and \ref{['theo:f-deltafloat1']}, and Propositions \ref{['corollary1']} and \ref{['corollary2']}
  12. Proof of Theorem \ref{['thm:sconcave']}

Key Result

Proposition 1

Let $\psi\in\text{Con}(\mathbb{R}^n)$ and $\Phi: \mathbb{R}^{n} \to \mathbb{R}_+$ be a continuous function.

Theorems & Definitions (29)

  • Remark 1
  • Definition 1
  • Proposition 1
  • Remark 2
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Remark 3
  • Proposition 2
  • Proposition 3
  • ...and 19 more