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

Chao Wang

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 persists for boxes subjected to small rotations and provide an explicit quantitative bound on the permissible rotation angle depending on the aspect ratio and the lattice gap. 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.

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

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 persists for boxes subjected to small rotations and provide an explicit quantitative bound on the permissible rotation angle depending on the aspect ratio and the lattice gap. 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 (7 sections, 5 theorems, 18 equations)

This paper contains 7 sections, 5 theorems, 18 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
  7. Sensitivity Analysis of Successive Minima

Key Result

Theorem 3.1

Let $K = \{x \in \mathbb{R}^d : |x_i| \le \alpha_i \}$ be an axis-aligned box with semi-axes $\alpha_1 \ge \dots \ge \alpha_d > 0$. Then Conjecture conj:main holds.

Theorems & Definitions (14)

  • Conjecture 1.1: Betke-Henk-Wills
  • Theorem 3.1: Conjecture for Boxes
  • proof
  • Remark 3.2
  • Proposition 4.1: Qualitative Local Stability
  • proof
  • Theorem 5.1: Explicit Stability Region
  • proof
  • Proposition 6.1: Stability of Successive Minima
  • proof
  • ...and 4 more