Connectivity when isolated vertices disappear in a convex polytope

Let $X_1,\ldots,X_n$ be independent uniform points in a fixed compact convex polytope $A\subset\R^d$ with nonempty interior, where $d\geq3$. Let $L_n$ be the largest nearest-neighbour distance and let $M_n$ be the connectivity radius of their Euclidean random geometric graph. We prove that $\Pp(L_n=M_n)\to1$. The argument retains the contribution of every face at a deterministic radius slightly below the leading connectivity scale. A linear gain in cone-ball volume excludes small components, and a positive collar surplus excludes medium separated sets. A fixed spatial net then excludes two large components simultaneously at every larger radius. An isolated vertex persists at the chosen lower radius, so these estimates apply at the random radius $L_n$.