The location of the most isolated point\ in a polyhedral sample

Let $A$ be a compact convex polyhedron in $\mathbb R^3$ with nonempty interior, and sample independent uniform points in $A$. We prove that the point with the largest nearest-neighbour distance converges in distribution to normalized surface area on $\partial A$ when every interior dihedral angle exceeds $\pi/2$. When the smallest angle is less than $\pi/2$, its limit is normalized length on the union of all edges with that angle. The proof computes spatial empty-ball intensities and joint exceedance probabilities for the fixed-size sample. The critical angle is excluded.