Computing intrinsic volumes of sublevel sets and applications

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 volumesofconvexpolynomial sublevel sets. More precisely, let f be a convex d–homogeneous polynomial of even degree d ≥ 2whichispositiveexceptattheorigin. Weshowthattheintrinsic/dualvolumesofthesublevelset[f ≤ 1] admit Laplace-type integral formulas obtained by averaging the infimal projection and restriction of f over the Grassmannian. This explicit representation yields:

(i) Löwner–John–type existence and uniqueness results, extending beyond the classical volume case; (ii) a block decomposition principle describing factorization of intrinsic volumes under direct-sum splitting; (iii) a coordinate-free formulation of Lipschitz-type lattice discrepancy bounds.

The resulting formulas enable analytic treatment for a broad class of geometric quantities, providing direct access to variational and arithmetic applications as well as new structural insights.