Computing intrinsic volumes of sublevel sets and applications
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914129669783552 |
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| author | Lê, Trí Minh Nguyen-Dang, Khai-Hoan |
| author_facet | Lê, Trí Minh Nguyen-Dang, Khai-Hoan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24001 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computing intrinsic volumes of sublevel sets and applications Lê, Trí Minh Nguyen-Dang, Khai-Hoan Metric Geometry Number Theory Optimization and Control 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. |
| title | Computing intrinsic volumes of sublevel sets and applications |
| topic | Metric Geometry Number Theory Optimization and Control |
| url | https://arxiv.org/abs/2510.24001 |