Local Stability and Quantitative Bounds for the Betke-Henk-Wills Conjecture

Abstract

The Betke-Henk-Wills conjecture provides an upper bound for the lattice point enumerator $G(K, \Lambda)$ of a convex body in terms of its successive minima. While the conjecture is established for orthogonal parallelotopes, its validity for general convex bodies in dimensions $d \ge 5$ remains open. In this paper, we examine the stability of the conjecture under metric perturbations. Specifically, we demonstrate that the inequality is strictly maintained for integer boxes subjected to rotations within a calculated radius, a consequence of the discrete nature of the lattice point enumerator. We derive explicit, geometry-invariant quantitative bounds on the perturbation radius using the operator norm. Furthermore, we extend the analysis to $L_p$-balls for sufficiently large $p$, identifying a sharp threshold $p_0$ for the invariance of the integer hull.