An Optimization Problem Related to Minkowski’s Successive Minima

Abstract

The purpose of this paper is to establish an inequality connecting the lattice point enumerator of a 0-symmetric convex body with its successive minima. To this end, we introduce an optimization problem whose solution refines former methods, thus producing a better upper bound. In particular, we show that an analogue of Minkowski's second theorem on successive minima with the volume replaced by lattice point enumerator is true up to an exponential factor, whose base is approximately 1.64.

A discrete analogue for Minkowski’s second theorem on successive minima

“…The estimates for C d in [5] yield Theorem 1.5. Notice that when K is not 0-symmetric, we cannot disregard all q i (K, Λ) that are less than or equal to 2, as it was done in [5]; the reason is that we may have q 1 (K, Λ) = · · · = q d (K, Λ) = 2, and K ∩ Λ may have full affine dimension (while if K was 0-symmetric, this would mean that K ∩ Λ = {0}).…”

Bounds on the Lattice Point Enumerator via Slices and Projections

“…which was later improved in [20]. The currently best known upper bound is due to Malikiosis [33], which, in particular, implies for K ∈ K n os that…”

Variations of Minkowski's theorem on successive minima

“…In Propositions 2.3 and 2.4, we verify this conjecture for the special cases n = 2 and simplices of arbitrary dimension, respectively. Another direction of extending Minkowski's 2nd theorem is to replace the volume functional by other functionals, for instance, the lattice point enumerator (see, e.g., [5,19,20]) or the intrinsic volumes (see, e.g., [15,28]). Here we are interested in inequalities analogous to (1.1) for the surface area.…”