Interesting statement about geometrical objects. Note that the unit hypercube (surface) aligned with the axis will not converge to a normal (regardless of orientation?), it stays uniform instead.

In fact the simplex seems to be the worst case for convex objects, in terms of concentration of distribution near the center. And the best case should be the sphere. Which plays out nicely since the simplex seems to be the most "concavey" convex shape of a given 'diameter' is the sphere is the most "convexey" convex shape of a given 'diameter', no?