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Monotonicity of the Gaussian measure under Banaszczyk transforms

Maud Szusterman

TL;DR

The work investigates monotonicity of the Gaussian measure under a Banaszczyk-type transform, introducing a Gaussian variant $K\circ u$ that cooperates with Ehrhard symmetrizations to simplify the original proof of the $5K$-theorem. By reducing the general monotonicity question to the planar half-plane case through a sequence of Ehrhard symmetrizations and isoperimetric inequalities, the authors isolate a core 2D problem and prove it for half-planes, thus yielding a streamlined path to the $5K$ bound. The paper also discusses conjectural generalizations, potential radius bounds for which monotonicity holds, and technical gaps resolved in the appendix, contributing to a clearer, more modular understanding of Banaszczyk-type monotonicity phenomena. Overall, the approach provides a conceptually simpler framework for bounding vector balancing constants via Gaussian-measure monotonicity and symmetrization techniques, with potential implications for algorithmic and geometric applications in high dimensions.

Abstract

In the proof of his famous 5K-theorem, W. Banaszczyk introduced a transformation of convex bodies for which the Gaussian measure is monotone. In this note, we present a simplified proof of this monotonicity by slightly modifying Banaszczyk's transform, so that it interacts smoothly with Ehrhard symmetrizations, thereby yielding a somewhat easier proof of the 5K-theorem.

Monotonicity of the Gaussian measure under Banaszczyk transforms

TL;DR

The work investigates monotonicity of the Gaussian measure under a Banaszczyk-type transform, introducing a Gaussian variant that cooperates with Ehrhard symmetrizations to simplify the original proof of the -theorem. By reducing the general monotonicity question to the planar half-plane case through a sequence of Ehrhard symmetrizations and isoperimetric inequalities, the authors isolate a core 2D problem and prove it for half-planes, thus yielding a streamlined path to the bound. The paper also discusses conjectural generalizations, potential radius bounds for which monotonicity holds, and technical gaps resolved in the appendix, contributing to a clearer, more modular understanding of Banaszczyk-type monotonicity phenomena. Overall, the approach provides a conceptually simpler framework for bounding vector balancing constants via Gaussian-measure monotonicity and symmetrization techniques, with potential implications for algorithmic and geometric applications in high dimensions.

Abstract

In the proof of his famous 5K-theorem, W. Banaszczyk introduced a transformation of convex bodies for which the Gaussian measure is monotone. In this note, we present a simplified proof of this monotonicity by slightly modifying Banaszczyk's transform, so that it interacts smoothly with Ehrhard symmetrizations, thereby yielding a somewhat easier proof of the 5K-theorem.

Paper Structure

This paper contains 7 sections, 7 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Reducing to the case of half-planes
  3. Proof that Lemma \ref{['halfplanelem']} implies Theorem \ref{['gaussianmono']}
  4. Monotonicity holds for half-planes
  5. Appendix: two remarks and a claim
  6. Remarks on conjectural variants of Theorem \ref{['gaussianmono']}
  7. Appendix : closing a small gap from Step (\ref{['eqn:step3']})

Key Result

Theorem 1.1

Let $K\subset \mathbb{R}^n$ be a closed convex set, $\gamma_n(K) \geq \frac{1}{2}$. Let $u_1, ... , u_t\in B_2^n$ (with $t\geq 1$ arbitrary). Then there exists $\epsilon\in \{\pm 1\}^t$ such that $\epsilon.u :=\epsilon_1 u_1+...+\epsilon_t u_t \in 5K$.

Figures (2)

  • Figure 1: Visualizing the halfplane $H$, and the cones $C_r(L)$ and $C_r(H)$, for Step 3.
  • Figure 2: A proof by picture of Claim \ref{['exoL1']}

Theorems & Definitions (12)

  • Theorem 1.1: Theorem 1, Ban98
  • Theorem 1.2: Theorem 3, Ban98
  • Theorem 1.3
  • Lemma 1.4
  • Proposition 1.5
  • Proposition 1.6: Theoreme 3.2, p. 292 ehrhard
  • Claim 2.1
  • proof
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['planarlem']}
  • ...and 2 more