A Generalization of the Discrete Version of Minkowski's Fundamental Theorem

Abstract

In memory of Hermann Minkowski on the occasion of his 150th birthday Abstract. One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of a 0-symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Our main contribution is a corresponding optimal relation between the number of boundary and interior lattice points of a 0-symmetric convex body. The proof relies on a congruence argument and a difference set estimate from additive combinatorics.

Tight bounds on discrete quantitative Helly numbers

“…(Here the strict convexity is a crucial assumption; see [20] for an analogous investigation without this assumption.) Thus, Corollary 6 shows that the study of c(Z n , k) can be viewed as a research in geometry of numbers dealing with the case of strictly convex bodies.…”

“…30.2] saying that every 0-symmetric strictly convex body with exactly one interior integer point contains at most 2 n+1 − 1 integer points in total. (Here the strict convexity is a crucial assumption; see [20] for an analogous investigation without this assumption.) Thus, Corollary 6 shows that the study of c(Z n , k) can be viewed as a research in geometry of numbers dealing with the case of strictly convex bodies.…”

Bounds on the Lattice Point Enumerator via Slices and Projections

“…In fact, if K ∈ K n os possesses 2h + 1 lattice points on the coordinate axis Re n , any interior lattice point v ∈ K ∩ e ⊥ n will contribute O v,n (h) lattice points to K . Here, O v,n hides a constant that only depends on v and n. However, unlike the simplex above, a symmetric convex body always contains at least #K /3 n interior lattice points (see [16]). Motivated by this heuristic, we conjecture the following polytopes to be extremal in (3.1), when restricted to K n os .…”