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In 1998, Richard Gardner and Gaoyong Zhang introduced the radial pth mean bodies $R_pK$ of a convex body K for $p>-1$. Furthermore, they established that $R_pK$ are convex when $p\ge 0$, but the convexity in the regime $(-1,0)$ remained unresolved. In this talk, we answer this nearly 30-year-old question in the affirmative. Along the way, we provide a new proof of Keith Ball's theorem on integrals of log-concave functions along rays against the weight $r^{p-1}$ and introduce an extension to negative $p$.