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On The Fourier Mean Bodies of a Convex Body

Dylan Langharst, Auttawich Manui, Artem Zvavitch

TL;DR

This paper develops a Fourier-analytic framework for radial $p$th mean bodies, introducing the Fourier $p$th mean bodies $F_pK$ and unveiling deep connections to classical objects in geometric tomography, such as intersection bodies and polar mean zonoids. It proves that $F_pK$ is an origin-symmetric $L^n$-star for $p\in(0,n)$, is convex for $p\in(0,1]$ (with $F_1K=\pi I(R_{n-1}K)$), but can be non-convex for $p>1$ (e.g., for cubes), and establishes affine-isoperimetric-type inequalities for $F_pK$ and its duals. The work also links $F_pK$ to $R_pK$, $Z_p^\circ K$, and $\Gamma_q^\circ K$, and derives isotropic bounds and asymptotic behavior in the isotropic setting, leveraging Berwald-type inequalities and recent slicing results. These results enrich the toolkit of geometric tomography by giving a Fourier-analytic lens on mean bodies, with implications for convexity, affine geometry, and stability under linear transformations.

Abstract

In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_p K$ of a convex body $K$ in $\mathbb{R}^n$ for $p>-1$, which now play an important role in geometric tomography. In this work, we study the Fourier transforms of the radial functions of $R_p K$. We introduce a new family of star-shaped sets $F_p K$, which we call the Fourier $p$th mean bodies of $K$. We are then interested in the convexity and the relevant affine-isoperimetric inequalities for $F_p K$, as well as connections of $F_p K$ with other classical objects in geometric tomography such as centroid bodies, intersection bodies, and mean zonoids. We also show that the bodies $F_p K$ are, for $p\in (0,1]$, close to ellipsoids in the sense of Hensley's theorem.

On The Fourier Mean Bodies of a Convex Body

TL;DR

This paper develops a Fourier-analytic framework for radial th mean bodies, introducing the Fourier th mean bodies and unveiling deep connections to classical objects in geometric tomography, such as intersection bodies and polar mean zonoids. It proves that is an origin-symmetric -star for , is convex for (with ), but can be non-convex for (e.g., for cubes), and establishes affine-isoperimetric-type inequalities for and its duals. The work also links to , , and , and derives isotropic bounds and asymptotic behavior in the isotropic setting, leveraging Berwald-type inequalities and recent slicing results. These results enrich the toolkit of geometric tomography by giving a Fourier-analytic lens on mean bodies, with implications for convexity, affine geometry, and stability under linear transformations.

Abstract

In 1998, R. Gardner and G. Zhang introduced the radial th mean bodies of a convex body in for , which now play an important role in geometric tomography. In this work, we study the Fourier transforms of the radial functions of . We introduce a new family of star-shaped sets , which we call the Fourier th mean bodies of . We are then interested in the convexity and the relevant affine-isoperimetric inequalities for , as well as connections of with other classical objects in geometric tomography such as centroid bodies, intersection bodies, and mean zonoids. We also show that the bodies are, for , close to ellipsoids in the sense of Hensley's theorem.
Paper Structure (23 sections, 45 theorems, 233 equations, 3 figures)

This paper contains 23 sections, 45 theorems, 233 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Intersection Bodies
  3. Radial Mean Bodies and Polar Mean Zonoids
  4. Fourier Mean Bodies
  5. Compactness and Convexity
  6. Isotropicity
  7. Affine isoperimetric-type inequalities
  8. Preparatory Facts
  9. Preliminaries
  10. On Intersection Bodies and Polar Centroid Bodies
  11. On $s$-concave Functions
  12. The Covariogram
  13. The Geometry of Radial Mean Bodies
  14. Basic properties
  15. Polar Mean Zonoids
  16. ...and 8 more sections

Key Result

Theorem A

Let $K\subset {\mathbb R}^n$ be a convex body. Then, for $p\geq 0$, $R_p K$ is an origin-symmetric convex body.

Figures (3)

  • Figure 1: The body $F_p K$ when $p \in (0,1]$ and $K=[-1,1]^2$. Convex.
  • Figure 2: The body $F_p K$ when $p=1.5$ and $p=2$; $K=[-1,1]^2$. Not Convex.
  • Figure 3: The body $R_p Q_2$ when $p=0.5$ and $p=1.5$.

Theorems & Definitions (82)

  • Theorem A: R. Gardner and G. Zhang GZ98
  • Theorem B: J. Haddad and M. Ludwig HL26
  • Theorem C: G. Berck BG09
  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Definition 5
  • Proposition 6
  • Theorem 7
  • ...and 72 more