Sharp phase transitions in Euclidean integral geometry

TL;DR

This paper derives refined concentration inequalities for intrinsic volumes of convex bodies, revealing new phase transitions in high-dimensional integral geometry related to random projections, slicing, and kinematic formulas.

Contribution

It introduces sharper concentration inequalities for intrinsic volumes, leading to the discovery of phase transitions in various high-dimensional geometric formulas.

Findings

Sharp concentration inequalities for intrinsic volumes.

Identification of phase transitions in high-dimensional geometric formulas.

Reduction of complex geometric interactions to single parameters.

Abstract

The intrinsic volumes of a convex body are fundamental invariants that capture information about the average volume of the projection of the convex body onto a random subspace of fixed dimension. The intrinsic volumes also play a central role in integral geometry formulas that describe how moving convex bodies interact. Recent work has demonstrated that the sequence of intrinsic volumes concentrates sharply around its centroid, which is called the central intrinsic volume. The purpose of this paper is to derive finer concentration inequalities for the intrinsic volumes and related sequences. These concentration results have striking implications for high-dimensional integral geometry. In particular, they uncover new phase transitions in formulas for random projections, rotation means, random slicing, and the kinematic formula. In each case, the location of the phase transition is determined by reducing each convex body to a single summary parameter.