Gardner–Giannopoulos cross-section body conjecture for $s$-concave densities

Let $f$ be a symmetric $s$-concave probability density on $\mathbb{R}^n$. For $p\in[-1,\infty]$, let $C_p(f)$ denote its $p$-th cross-section body, defined through its radial function by

$$\rho_{C_p(f)}(v)=\left(\int_{\mathbb{R}^n}\left(\int_{x+v^\perp}f(y)\,dy\right)^p f(x)\,dx\right)^{1/p},\qquad v\in\mathbb{S}^{n-1}.$$

Cross-section body conjecture. If $s\in[-1/n,\infty]$, then $C_p(f)$ is a symmetric convex body. This is the geometric reformulation of the reverse Rényi entropy power norm conjecture. It extends the intersection-body and cross-section-body phenomena known for symmetric convex bodies and, in some cases, symmetric log-concave or convex measures; the general assertion remains open in the source.