Convexity of Radial Mean Bodies via an extension of Ball's Bodies

In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight $r^{p-1}$ to the previously inaccessible regime $p\in (-1,0)$: if $g:{\mathbb{R}}^n\to {\mathbb{R}}_{+}$ is an integrable log-concave function which attains its maximum at the origin, then $$x\mapsto {(\frac{p}{g(o)}{\int }_0^{\mathrm{\infty }}{r}^{p-1}(g(rx)-g(o))\mathrm{d}r)}^{-\frac{1}{p}}$$ is a positively 1-homogeneous convex function on ${\mathbb{R}}^n$. Our approach also provides a new proof of the original regime $p>0$. The argument is based on a reduction to a two-dimensional inequality derived from Prékopa’s theorem, which may be of independent interest. As a consequence of this extension, we resolve a nearly 30-year-old question of Richard Gardner and Gaoyong Zhang in the affirmative. In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_{p}K$ of a convex body $K\subset {\mathbb{R}}^n$ for $p>-1$. Furthermore, they established that $R_{p}K$ is convex for $p\ge 0$, but the convexity of $R_{p}K$ for $p\in (-1,0)$ remained open. We prove that $R_{p}K$ is convex for all $p>-1$.