On the Fourier Mean Bodies of a Convex Body

Abstract

In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_{p}K$ of a convex body $K\subset {\mathbb{R}}^n$, $p>-1$, which have since become important objects in geometric tomography. In this paper we study the Fourier transforms of the radial functions of $R_{p}K$. This leads to a new family of star-shaped sets $F_{p}K$, which we call the Fourier $p$th mean bodies of $K$. We prove Fourier inversion formulas connecting $R_{p}K$ and $F_{p}K$, realizing them as $p$-intersection bodies in the sense of A. Koldobsky. We develop the basic affine geometry of $F_{p}K$; this includes affine invariance and monotonicity properties. We identify the range of $p$ where $F_{p}K$ is compact in terms of the decay of $|\widehat{{\chi }_K}{|}^2$. We show that $F_{p}K$ is an origin-symmetric convex body for every $0<p\le 1$. This range is sharp in general: already for the cube, $F_p[-1,1{]}^n$ is not convex for $1<p<2$ and $n\ge 2,$ while $F_p[-1,1{]}^n$ is not compact for $p\ge 2$. We further investigate the features Fourier mean bodies share with intersection bodies: we prove Hensley-type estimates for $F_{p}K$ when $K$ is isotropic and investigate a few affine isoperimetric inequalities.

six pictures; v2: Improved Hensley-type theorem, rewrote introduction and abstract and switched to default letter margins (the latter change pushed the paper from 53 pages to 74 pages)