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 $n=2.$
EDIT There seem to be confusion about what I mean by "the length of inward pointing normal": Take the point $x.$ There is a line $l$ through $x$ normal to $S.$ Since $S$ is convex, $l$ intersects $S$ twice. The distance between the two intersection points is $s(x).$
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 $n=2.$