On the Fourier Mean Bodies of a Convex Body

In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_pK$ 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 $R_pK$. This leads to a new family of star-shaped sets $F_pK$, which we call the Fourier $p$th mean bodies of $K$. We prove Fourier inversion formulas connecting $R_pK$ and $F_pK$, realizing them as $p$-intersection bodies in the sense of A. Koldobsky. We develop the basic affine geometry of $F_pK$; 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 $F_pK$ is an origin-symmetric convex body for every $0<p\le1$. This range is sharp in general: already for the cube, $F_p[-1,1]^n$ is not convex for $1<p<2$ and $n\geq 2,$ while $F_p[-1,1]^n$ is not compact for $p\geq 2$. We further investigate the features Fourier mean bodies share with intersection bodies: we prove Hensley-type estimates for $F_pK$ when $K$ is isotropic and investigate a few affine isoperimetric inequalities.