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Extremal sections and projections of certain convex bodies: a survey

Piotr Nayar, Tomasz Tkocz

TL;DR

The survey synthesizes sharp bounds for volumes of sections and projections of convex bodies, emphasizing the cube, $\ell_p$-balls, and simplices, and highlights a cohesive probabilistic-Fourier framework built on $L_p$–inequalities, isotropic position, and Brascamp–Lieb theory. It catalogs classical results (e.g., Vaaler, Hadwiger–Hensley, Ball) and modern extensions, including phase transitions for $B_p^n$ and complex-analytic analogues, while connecting these geometric questions to entropy-power inequalities and the logarithmic Brunn–Minkowski conjecture. The work emphasizes how majorisation, Gaussian mixtures, and Schur-convexity underpin extremal problems for sections and projections, and it outlines 11 conjectures that guide ongoing research. Overall, the paper builds a unified, multidimensional toolkit for extremal volumetric questions in high-dimensional convex geometry, with implications for analysis, probability, and geometric functional analysis.

Abstract

We survey results concerning sharp estimates on volumes of sections and projections of certain convex bodies, mainly $\ell_p$ balls, by and onto lower dimensional subspaces. This subject emerged from geometry of numbers several decades ago and since then has seen development of a variety of probabilistic and analytic methods, showcased in this survey.

Extremal sections and projections of certain convex bodies: a survey

TL;DR

The survey synthesizes sharp bounds for volumes of sections and projections of convex bodies, emphasizing the cube, -balls, and simplices, and highlights a cohesive probabilistic-Fourier framework built on –inequalities, isotropic position, and Brascamp–Lieb theory. It catalogs classical results (e.g., Vaaler, Hadwiger–Hensley, Ball) and modern extensions, including phase transitions for and complex-analytic analogues, while connecting these geometric questions to entropy-power inequalities and the logarithmic Brunn–Minkowski conjecture. The work emphasizes how majorisation, Gaussian mixtures, and Schur-convexity underpin extremal problems for sections and projections, and it outlines 11 conjectures that guide ongoing research. Overall, the paper builds a unified, multidimensional toolkit for extremal volumetric questions in high-dimensional convex geometry, with implications for analysis, probability, and geometric functional analysis.

Abstract

We survey results concerning sharp estimates on volumes of sections and projections of certain convex bodies, mainly balls, by and onto lower dimensional subspaces. This subject emerged from geometry of numbers several decades ago and since then has seen development of a variety of probabilistic and analytic methods, showcased in this survey.
Paper Structure (22 sections, 46 theorems, 156 equations, 2 tables)

This paper contains 22 sections, 46 theorems, 156 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Prologue
  3. The motivating example
  4. Preliminaries and overview
  5. Existing literature and our aim
  6. Sections
  7. Cube
  8. Balls of $p$-norms
  9. Simplex
  10. Complex analogues
  11. Miscellanea
  12. Projections
  13. Cube
  14. Simplex
  15. Cross-polytope
  16. ...and 7 more sections

Key Result

Theorem 1

Let $K$ be a symmetric convex body in $\mathbb{R}^n$. Then the function extended by $0$ at $x=0$ defines a norm on $\mathbb{R}^n$.

Theorems & Definitions (76)

  • Theorem 1: Busemann Bus
  • Theorem 2: Cauchy-Minkowski
  • Theorem 3: Hensley Hen-gen
  • Theorem 4: Hadwiger Hadw, Hensley Hen
  • proof
  • Theorem 5: Vaaler Vaa
  • Theorem 6: Ball Ball
  • proof : Sketch of the proof
  • Theorem 7: Ball Ball2
  • Conjecture 1
  • ...and 66 more