{"ID":2848867,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.24001","arxiv_id":"2510.24001","title":"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 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) Löwner--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.","short_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...","url_abs":"https://arxiv.org/abs/2510.24001","url_pdf":"https://arxiv.org/pdf/2510.24001v2","authors":"[\"Trí Minh Lê\",\"Khai-Hoan Nguyen-Dang\"]","published":"2025-10-28T02:15:43Z","proceeding":"math.MG","tasks":"[\"math.MG\",\"math.NT\",\"math.OC\"]","methods":"[]","has_code":false}