$f:\mathbf{R}^d\to \mathbf{R}_{\ge 0}$ is log-concave if $\log(f)$ is concave (and the domain of $\log(f)$ is convex).
Theorem: For all $\sigma$ on the sphere $\Bbb S^{d-1}$ and $r\in \mathbf{R}$, $$ g_\sigma(r) := \int\limits_{\sigma\cdot x=r}f(x)\,\mathrm{d}S(x) $$ is a log-concave function of $r$. (Note: $g$, as a function of $\sigma$ and $r$, is the Radon transform of $f$.)
Question: does this characterize log-concavity? That is, if $g_\sigma(r)$ is log-concave as a function of $r$ for all $\sigma$, is $f$ log-concave?