Computing intrinsic volumes of sublevel sets and applications

TL;DR

This paper introduces a unified integral formula for intrinsic volumes of convex polynomial sublevel sets, leading to new geometric insights, existence results, and decomposition principles.

Contribution

It provides a novel Laplace-Grassmannian integral representation for intrinsic and dual volumes of convex polynomial sublevel sets, extending classical geometric results.

Findings

Derived explicit Laplace-type integral formulas for intrinsic volumes

Established L"owner--John-type existence and uniqueness results

Developed a block decomposition principle for intrinsic volumes

Abstract

Intrinsic volumes are fundamental geometric invariants generalizing volume, surface area, and mean width for convex bodies. We establish a unified Laplace-Grassmannian representation for intrinsic and dual volumes of convex polynomial sublevel sets. More precisely, let $f$ be a convex $d$-homogeneous polynomial of even degree $d \ge 2$ which is positive except at the origin. We show that the intrinsic and dual volumes of the sublevel set $[f \le 1]$ admit Laplace-type integral formulas obtained by averaging the infimal projection and restriction of $f$ over the Grassmannian. This explicit representation yields three main consequences: (1) L\"owner--John-type existence and uniqueness results extending beyond the classical volume case; (2) a block decomposition principle describing factorization of intrinsic volumes under direct-sum splitting; (3) a coordinate-free formulation of Lipschitz-type lattice discrepancy bounds. These formulas enable analytic treatment of a broad class of geometric quantities, providing direct access to variational and arithmetic applications as well as new structural insights.