Inverse formula for the Blaschke-Levy representation with applications to zonoids and sections of star bodies

Abstract

We say that an even continuous function $H$ on the unit sphere $\Omega$ in $R^n$ admits the Blaschke-Levy representation with $q>0$ if there exists an even function $b\in L_1(\Omega)$ so that $H^q(x)=\int_\Omega |(x,\xi)|^q b(\xi)\ d\xi$ for every $x\in \Omega.$ This representation has numerous applications in convex geometry, probability and Banach space theory. In this paper, we present a simple formula (in terms of the derivatives of $H$) for calculating $b$ out of $H.$ We use this formula to give a sufficient condition for isometric embedding of a space into $L_p$ which contributes to the 1937 P.Levy's problem and to the study of zonoids. Another application gives a Fourier transform formula for the volume of $(n-1)$-dimensional central sections of star bodies in $R^n.$ We apply this formula to find the minimal and maximal volume of central sections of the unit balls of the spaces $\ell_p^n$ with $0<p<2.$