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 $K$ 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.