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Quantitative Stability of the Betke-Henk-Wills Conjecture

Chao Wang

TL;DR

Addresses the Betke-Henk-Wills conjecture that bounds the lattice point count $G(K, \Lambda)$ by a product involving the successive minima $\lambda_i(K, \Lambda)$. The authors prove strict local stability for integer boxes under small rotations and derive explicit operator-norm–based perturbation bounds, and they extend stability to large-$p$ deformations of $L_p$ balls via a sharp threshold $p_0$. The approach combines Lipschitz continuity of $\lambda_i$ with discrete lattice counting to control entering lattice points and uses a boundary-distance argument to obtain quantitative stability. Overall, the results demonstrate robustness of the conjecture under perturbations and connect discrete lattice geometry with continuous deformations, with potential implications for Banach-space geometry and the geometry of numbers.

Abstract

The Betke-Henk-Wills conjecture proposes a sharp upper bound for the lattice point enumerator $G(K, Λ)$ 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.

Quantitative Stability of the Betke-Henk-Wills Conjecture

TL;DR

Addresses the Betke-Henk-Wills conjecture that bounds the lattice point count by a product involving the successive minima . The authors prove strict local stability for integer boxes under small rotations and derive explicit operator-norm–based perturbation bounds, and they extend stability to large- deformations of balls via a sharp threshold . The approach combines Lipschitz continuity of with discrete lattice counting to control entering lattice points and uses a boundary-distance argument to obtain quantitative stability. Overall, the results demonstrate robustness of the conjecture under perturbations and connect discrete lattice geometry with continuous deformations, with potential implications for Banach-space geometry and the geometry of numbers.

Abstract

The Betke-Henk-Wills conjecture proposes a sharp upper bound for the lattice point enumerator of a convex body in terms of its successive minima. While the conjecture remains open for general convex bodies in dimensions , 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 -balls for sufficiently large , deriving a sharp threshold for the stability of the integer hull.
Paper Structure (6 sections, 5 theorems, 19 equations)

This paper contains 6 sections, 5 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. The Baseline Case: Orthogonal Parallelotopes
  4. Local Stability under Rotation
  5. Quantitative Stability Bounds
  6. Asymptotic Stability for $L_p$ Deformations

Key Result

Lemma 2

Let $K \in \mathcal{K}^d_0$. For any non-singular linear transformation $T \in GL(d, \mathbb{R})$, the successive minima satisfy:

Theorems & Definitions (12)

  • Conjecture 1: Betke-Henk-Wills
  • Lemma 2: Lipschitz Continuity of Successive Minima
  • proof
  • Theorem 3: Conjecture for Boxes
  • proof
  • Remark 4
  • Theorem 5: Strict Stability in Critical Configurations
  • proof
  • Theorem 6: Explicit Stability Radius
  • proof
  • ...and 2 more