Optimal Arrangement of Finite Point Sets in Riemannian 2-Manifolds

First a stability version of a theorem of L. Fejes Tóth on sums of moments is given: a large finite point set in a $2$-dimensional Riemannian manifold, for which a certain sum of moments is minimal, must be an approximately regular hexagonal pattern. This result is then applied to show the following: (i) The nodes of optimal numerical integration formulae for Hoelder continuous functions on such manifolds form approximately regular hexagonal patterns if the number of nodes is large. (ii) Given a smooth convex body in $\mathbb E^3$, most facets of the circumscribed convex polytopes of minimum volume in essence are affine regular hexagons if the number of facets is large. A similar result holds with volume replaced by mean width. (iii) A convex polytope in $\mathbb E^3$ of minimal surface area, amongst those of given volume and given number of facets, has the property that most of its facets are almost regular hexagons assuming the number of facets is large.