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Empirical forms of the Petty projection inequality

Grigoris Paouris, Peter Pivovarov, Kateryna Tatarko

TL;DR

This work introduces stochastic (empirical) forms of the Petty projection inequality by examining random linear images of convex bodies and randomized centroid/projection constructions. It establishes sharp, high-signal inequalities for expected values of mixed projection bodies and mixed volumes, via rearrangement and shadow-system techniques, linking projection bodies, centroid bodies, and Minkowski-type operations. The authors derive multiple empirical inequalities, including for measures and mixed projections, and show convergence of empirical constructions to deterministic centroids, from which Petty’s inequality can be recovered in the large-sample limit. The results deepen the affine isoperimetric landscape by providing stochastic analogs and limits that illuminate the roles of symmetry and rearrangement in projection- and centroid-based inequalities, with potential implications for Brunn–Minkowski theory and related valuations. The methods unify random matrices, $L_p$- and $M$-additions, and shadow-system convexity to yield a suite of extremal, empirically sharp inequalities for random convex sets.

Abstract

The Petty projection inequality is a fundamental affine isoperimetric principle for convex sets. It has shaped several directions of research in convex geometry which forged new connections between projection bodies, centroid bodies, and mixed volume inequalities. We establish several different empirical forms of the Petty projection inequality by re-examining these key relationships from a stochastic perspective. In particular, we derive sharp extremal inequalities for several multiple-entry functionals of random convex sets, including mixed projection bodies and mixed volumes.

Empirical forms of the Petty projection inequality

TL;DR

This work introduces stochastic (empirical) forms of the Petty projection inequality by examining random linear images of convex bodies and randomized centroid/projection constructions. It establishes sharp, high-signal inequalities for expected values of mixed projection bodies and mixed volumes, via rearrangement and shadow-system techniques, linking projection bodies, centroid bodies, and Minkowski-type operations. The authors derive multiple empirical inequalities, including for measures and mixed projections, and show convergence of empirical constructions to deterministic centroids, from which Petty’s inequality can be recovered in the large-sample limit. The results deepen the affine isoperimetric landscape by providing stochastic analogs and limits that illuminate the roles of symmetry and rearrangement in projection- and centroid-based inequalities, with potential implications for Brunn–Minkowski theory and related valuations. The methods unify random matrices, - and -additions, and shadow-system convexity to yield a suite of extremal, empirically sharp inequalities for random convex sets.

Abstract

The Petty projection inequality is a fundamental affine isoperimetric principle for convex sets. It has shaped several directions of research in convex geometry which forged new connections between projection bodies, centroid bodies, and mixed volume inequalities. We establish several different empirical forms of the Petty projection inequality by re-examining these key relationships from a stochastic perspective. In particular, we derive sharp extremal inequalities for several multiple-entry functionals of random convex sets, including mixed projection bodies and mixed volumes.
Paper Structure (15 sections, 9 theorems, 93 equations, 1 table)

This paper contains 15 sections, 9 theorems, 93 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Petty projection inequality and its reach
  3. Empirical mixed projection body inequalities
  4. Empirical mixed volume inequalities
  5. Preliminaries
  6. Convex bodies and mixed volumes
  7. $L_p$ and $M$-addition operations
  8. Symmetrization of sets and functions
  9. Convexity and rearrangement inequalities
  10. Shadow systems
  11. Analytic tools
  12. Minimizing the mixed volume of random convex sets
  13. An empirical Petty projection inequality for measures
  14. Mixed projection bodies
  15. Laws of large numbers and convergence

Key Result

Theorem 1.1

Let $C_1,\ldots,C_{n-1}$ be compact convex sets such that $\mathop{\rm dim}(C_i)=m_i$ for $i=1,\ldots,n-1$ and let $m=m_1+\ldots+m_{n-1}$. Let $\bm{X}$ and $\bm{X}^{\#}$ be $n\times m$ random matrices with $\ell=n-1$ in Table table::notation. Then for any radial measure $\nu$ with a decreasing densi

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 3.6
  • Theorem 3.7
  • Corollary 5.8
  • Remark 5.9
  • Proposition 7.10
  • ...and 1 more