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On Sections of Convex Bodies in John's Position and of Generalised $B_p^n$ Balls

David Alonso-Gutiérrez, Silouanos Brazitikos, Giorgos Chasapis

TL;DR

This work develops sharp, John-position–based bounds for volumes and mean widths of k-dimensional sections of convex bodies. By marrying a Parseval-type identity with Brascamp–Lieb inequalities, the authors circumvent the lack of coordinate product structure and generalize Ball’s cube slicing results to arbitrary John decompositions, including non-symmetric bodies. They extend the analysis to generalized ell_p^n balls, providing explicit bounds for K_p ∩ H in terms of the John data and the one-dimensional Fourier transforms γ_p, and they use the Wills functional to obtain alternative proofs and mean-width results. The results reveal a threshold phenomenon: in symmetric cases, bounds improve under projection-size conditions, and in the non-symmetric setting, improved hyperplane-section bounds emerge, broadening the toolbox for geometric tomography and high-dimensional convex geometry.

Abstract

We revisit an ingenious argument of K. Ball to provide sharp estimates for the volume of sections of a convex body in John's position. Our technique combines the geometric Brascamp-Lieb inequality with a generalised Parseval-type identity. This lets us complement some earlier results of the first two named authors, as well as generalise the classical estimates of Meyer-Pajor and Koldobsky regarding extremal sections of $B_p^n$ balls to a broader family of norms induced by a John's decomposition of the identity in $\mathbb{R}^n$.

On Sections of Convex Bodies in John's Position and of Generalised $B_p^n$ Balls

TL;DR

This work develops sharp, John-position–based bounds for volumes and mean widths of k-dimensional sections of convex bodies. By marrying a Parseval-type identity with Brascamp–Lieb inequalities, the authors circumvent the lack of coordinate product structure and generalize Ball’s cube slicing results to arbitrary John decompositions, including non-symmetric bodies. They extend the analysis to generalized ell_p^n balls, providing explicit bounds for K_p ∩ H in terms of the John data and the one-dimensional Fourier transforms γ_p, and they use the Wills functional to obtain alternative proofs and mean-width results. The results reveal a threshold phenomenon: in symmetric cases, bounds improve under projection-size conditions, and in the non-symmetric setting, improved hyperplane-section bounds emerge, broadening the toolbox for geometric tomography and high-dimensional convex geometry.

Abstract

We revisit an ingenious argument of K. Ball to provide sharp estimates for the volume of sections of a convex body in John's position. Our technique combines the geometric Brascamp-Lieb inequality with a generalised Parseval-type identity. This lets us complement some earlier results of the first two named authors, as well as generalise the classical estimates of Meyer-Pajor and Koldobsky regarding extremal sections of balls to a broader family of norms induced by a John's decomposition of the identity in .
Paper Structure (12 sections, 26 theorems, 237 equations)

This paper contains 12 sections, 26 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Parseval's identity and consequences
  3. Centrally Symmetric Case
  4. A comparison between estimates
  5. The Wills functional of sections of centrally symmetrc convex bodies in John's position
  6. The volume of sections via the Wills functional
  7. The mean width of sections via the Wills functional
  8. Sections of Generalised $\ell_p^n$ Balls
  9. Sections of $K_1$
  10. Sections of $K_p$, $p\in[1,2]$
  11. Non-Symmetric Case
  12. Parseval's identity for characteristic functions

Key Result

Proposition 1

Let $m\in\mathbb{N}$, $n_1,\ldots,n_m\in\mathbb{N}$ and set $N=n_1+\ldots+n_m$. For every linear subspace $H\in G_{N,k}$ and every family of functions $(f_j)_{j=1}^m$ such that $f_j\in{\mathcal{S}}(\mathbb{R}^{n_j})$, $j=1,\ldots,m$,

Theorems & Definitions (56)

  • Proposition 1: Parseval
  • proof
  • Proposition 2
  • proof
  • Corollary 3
  • proof
  • Remark 4
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 46 more