Quantitative Stability of the Betke-Henk-Wills Conjecture

TL;DR

This paper investigates the stability of the Betke-Henk-Wills conjecture for convex bodies, proving it remains valid under small perturbations for certain shapes and extending its applicability to large p-value Lp-balls.

Contribution

It establishes the local stability of the conjecture for integer boxes under small rotations and extends the conjecture's validity to large p-value Lp-balls.

Findings

Proved strict stability for integer boxes with small rotations.

Derived explicit bounds on perturbation radius using operator norm.

Extended the conjecture's validity to Lp-balls with large p, identifying a sharp threshold p0.

Abstract

The Betke-Henk-Wills conjecture proposes a sharp upper bound for the lattice point enumerator $G(K, \Lambda)$ of a convex body in terms of its successive minima. While the conjecture remains open for general convex bodies in dimensions $d \ge 5$, it is known to hold for orthogonal parallelotopes (boxes). In this paper, we establish the \textit{local stability} of the conjecture under small perturbations of the metric. Specifically, we prove that the inequality is strictly stable for integer boxes subjected to small rotations, owing to the discrete nature of the lattice point counting function. We derive explicit, geometry-invariant quantitative bounds on the permissible perturbation radius using the operator norm. Furthermore, we extend the validity of the conjecture to a class of $L_p$-balls for sufficiently large $p$, deriving a sharp threshold $p_0$ for the stability of the integer hull.