f:**R**^{d}→**R**_{≥0} is *log-concave* if log(f) is concave (and the domain of log(f) is convex).

Theorem: For all σ on the sphere S^{d-1} and r∈**R**, g_{σ}(r) := ∫_{σ.x=r}f(x)dS(x) is a log-concave function of r. (Note: g, as a function of σ and r, is the Radon transform of f.)

Question: does this characterize log-concavity? That is, if g_{σ}(r) is log-concave as a function of r for all σ, is f log-concave?