On The Fourier Mean Bodies of a Convex Body

TL;DR

This paper explores the Fourier transforms of radial functions of radial pth mean bodies of convex bodies, introduces Fourier pth mean bodies, and investigates their convexity, inequalities, and relations to classical geometric objects.

Contribution

It introduces the Fourier pth mean bodies of convex bodies and studies their properties, convexity, and connections with classical objects in geometric tomography.

Findings

Fourier pth mean bodies are close to ellipsoids for p in (0,1].

Established affine-isoperimetric inequalities for these bodies.

Connected Fourier pth mean bodies with centroid bodies, intersection bodies, and mean zonoids.

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.