OAI.CoveringOrder.optimal_order

The theorem states that the proposition OptimalOrder holds, i.e. there exist real constants c>0 and C>0 and a threshold n₀ such that for every dimension n≥n₀ four extremal covering-density quantities each lie, as extended nonnegative reals, between c·n·log n and C·n·log n (so each is in particular finite). The quantities are suprema over convex bodies in n-dimensional Euclidean space, where a convex body is a compact convex set with nonempty interior. The first is the supremum over all convex bodies K of the translative covering density thetaT(K), the infimum of vol(K) times the upper center intensity over locally finite center sets X for which the translates K+x, x in X, cover the space; the upper center intensity is the limsup as R→∞ of the number of centers in the cube [−R,R]^n divided by (2R)^n. The second is the same supremum restricted to centrally symmetric convex bodies, meaning those satisfying x∈K iff 2z−x∈K for some center z. The third is the supremum over all convex bodies of the lattice covering density thetaL(K), the infimum over discrete full-rank lattices Λ with Λ+K covering the space of vol(K) divided by the covolume of Λ. The fourth is that lattice supremum restricted to centrally symmetric convex bodies.