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

TL;DR

This paper investigates the stability of the Betke-Henk-Wills conjecture for convex bodies under perturbations, providing explicit bounds and extending results to large p-norm balls, thus advancing understanding of lattice point enumeration bounds.

Contribution

It establishes quantitative stability bounds for the conjecture under metric perturbations and extends the analysis to large p-norm balls, identifying a sharp threshold for invariance.

Findings

The inequality holds for integer boxes under rotations within a specific radius.

Explicit bounds on perturbation radius are derived using the operator norm.

The analysis extends to L_p-balls for large p, with a sharp threshold identified.

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.