From Steiner Formulas for Cones to Concentration of Intrinsic Volumes

TL;DR

This paper introduces a probabilistic approach using a Steiner formula for cones to analyze intrinsic volumes, providing new identities, bounds, and a concentration inequality relevant for understanding random convex optimization problems.

Contribution

It develops a systematic probability-based method for studying conic intrinsic volumes, connecting geometric functionals with Gaussian analysis.

Findings

New identities for intrinsic volumes

Bounds for cone intrinsic volumes

Near-optimal concentration inequality

Abstract

The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random convex optimization problems. This paper develops a systematic technique for studying conic intrinsic volumes using methods from probability. At the heart of this approach is a general Steiner formula for cones. This result converts questions about the intrinsic volumes into questions about the projection of a Gaussian random vector onto the cone, which can then be resolved using tools from Gaussian analysis. The approach leads to new identities and bounds for the intrinsic volumes of a cone, including a near-optimal concentration inequality.