Given a convex surface $S$ in $\mathbb{R}^3,$ you can associate to each point the length of the inward pointing normal contained inside $S.$ Call this quantity $s(x).$ Now, given a positive function $f$ on the sphere $S^2,$ are necessary and sufficient conditions known so that $f$ can be realized as $s$ for some convex realization? There is the obvious uniqueness question, as well. The question can be formulated identically in $\mathbb{R}^n,$ but these sort of questions tend to get harder as $n$ gets larger. On the other hand, it is not obvious what the answer is even for $n=2.$