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Stability of polydisc slicing

Nathaniel Glover, Tomasz Tkocz, Katarzyna Wyczesany

TL;DR

This paper extends Ball's cube slicing results to the complex setting of polydiscs and proves a dimension-free stability theorem for the normalized section function $A_n(a)$, which bounds the volume of polydisc hyperplane sections. The authors reveal an extra asymptotic extremiser at the uniform coordinate vector and establish a dual bound that combines a geometric deficit term $|a-e_1|^2$ with a Fourier-analytic deficit measured by the $\ell_4$-norm, reflecting proximity to both extremisers. The proof blends probabilistic self-improvement near extremisers, Fourier-analytic estimates via a Bessel-function integral inequality, a Lipschitz/complex-intersection-body framework, and a Berry-Esseen bound to handle Gaussian-approximation regimes. A key outcome is a dimension-free, two-fold stability that tightens the complex analogue of Ball's inequality and highlights a Gaussian asymptotic extremiser as $n\to\infty$, enriching the real-case picture with new phenomena and techniques.

Abstract

We prove a dimension-free stability result for polydisc slicing due to Oleszkiewicz and Pelczyński (2000). Intriguingly, compared to the real case, there is an additional asymptotic maximiser. In addition to Fourier-analytic bounds, we crucially rely on a self-improving feature of polydisc slicing, established via probabilistic arguments.

Stability of polydisc slicing

TL;DR

This paper extends Ball's cube slicing results to the complex setting of polydiscs and proves a dimension-free stability theorem for the normalized section function , which bounds the volume of polydisc hyperplane sections. The authors reveal an extra asymptotic extremiser at the uniform coordinate vector and establish a dual bound that combines a geometric deficit term with a Fourier-analytic deficit measured by the -norm, reflecting proximity to both extremisers. The proof blends probabilistic self-improvement near extremisers, Fourier-analytic estimates via a Bessel-function integral inequality, a Lipschitz/complex-intersection-body framework, and a Berry-Esseen bound to handle Gaussian-approximation regimes. A key outcome is a dimension-free, two-fold stability that tightens the complex analogue of Ball's inequality and highlights a Gaussian asymptotic extremiser as , enriching the real-case picture with new phenomena and techniques.

Abstract

We prove a dimension-free stability result for polydisc slicing due to Oleszkiewicz and Pelczyński (2000). Intriguingly, compared to the real case, there is an additional asymptotic maximiser. In addition to Fourier-analytic bounds, we crucially rely on a self-improving feature of polydisc slicing, established via probabilistic arguments.
Paper Structure (16 sections, 13 theorems, 67 equations, 1 figure)

This paper contains 16 sections, 13 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A sketch of our approach
  3. Ancillary results and tools
  4. The role of independence
  5. Integral inequality
  6. Lipschitz property of the section function and complex intersection bodies
  7. Berry-Esseen bound
  8. Proof of Theorem \ref{['thm:main']}
  9. Two largest coordinates are close to 1/sqrt(2): local stability via self-improvement
  10. All weights are small
  11. Largest weight is moderately below 1/sqrt(2)
  12. Second largest weight is small
  13. Second largest weight is bounded away from $0$
  14. Largest weight is moderately above 1/sqrt(2)
  15. Largest weight is bounded below away from 1/sqrt(2)
  16. ...and 1 more sections

Key Result

Theorem 1

For $n \geq 2$ and every unit vector $a$ in $\mathbb{R}^n$ with $a_1 \geq a_2 \geq \dots \geq a_n \geq 0$, we have as well as

Figures (1)

  • Figure 1: We consider six cases. The labels L$k$ correspond to the lemmas in which a given case is resolved. In Section \ref{['sub:self-impr']} we explain the case where two largest coordinates are near $\tfrac{1}{\sqrt{2}}$, corresponding to L7 in the picture above. In Section \ref{['sub:all_small']} we explain the bound when all cooridinates are below $\sqrt{3/8}$, i.e. we cover the region L8. In Section \ref{['sub:below1root2']} we study the case where $a_1$ is below $1/\sqrt{2}$, which we examine in two regimes depending on the value of $a_2$ corresponding to L9 and L10. We address the case when $a_1$ is only slightly above $\frac{1}{\sqrt{2}}$, marked as L12, in Section \ref{['sub:a1_just_above']}. Finally, in Section \ref{['sub:a1_large']} we complete the picture by settling the case when $a_1$ is large ( L13). We put these bounds together, proving the theorem, in Section \ref{['sub:proof_main']}.

Theorems & Definitions (26)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 2
  • proof : Proof of Lemma \ref{['lm:X+Y']}
  • Lemma 3
  • proof
  • Theorem 4: Koldobsky-Paouris-Zymonopoulou, KPZ
  • Lemma 5
  • ...and 16 more