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Computing intrinsic volumes of sublevel sets and applications

Trí Minh Lê, Khai-Hoan Nguyen-Dang

TL;DR

This work advances intrinsic and dual intrinsic volumes for convex polynomial sublevel sets by deriving a unified Laplace–Grassmannian representation that averages the infimal projection and restriction of a convex, even-degree homogeneous polynomial over Grassmannians. This leads to Löwner–John–type existence/uniqueness results beyond ellipsoids, a block-decomposition principle for direct-sum splits, and a coordinate-free Lipschitz-type lattice-discrepancy framework. The results bridge convex geometry with arithmetic applications, providing explicit, computable representations that yield structural insights and enable lattice counting and theta-function analyses via intrinsic volumes. Their approach unifies variational methods, integral geometry, and lattice counting, with potential impact on optimization, number theory, and high-dimensional geometry.

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.

Computing intrinsic volumes of sublevel sets and applications

TL;DR

This work advances intrinsic and dual intrinsic volumes for convex polynomial sublevel sets by deriving a unified Laplace–Grassmannian representation that averages the infimal projection and restriction of a convex, even-degree homogeneous polynomial over Grassmannians. This leads to Löwner–John–type existence/uniqueness results beyond ellipsoids, a block-decomposition principle for direct-sum splits, and a coordinate-free Lipschitz-type lattice-discrepancy framework. The results bridge convex geometry with arithmetic applications, providing explicit, computable representations that yield structural insights and enable lattice counting and theta-function analyses via intrinsic volumes. Their approach unifies variational methods, integral geometry, and lattice counting, with potential impact on optimization, number theory, and high-dimensional geometry.

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 be a convex -homogeneous polynomial of even degree which is positive except at the origin. We show that the intrinsic and dual volumes of the sublevel set admit Laplace-type integral formulas obtained by averaging the infimal projection and restriction of 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.

Paper Structure

This paper contains 14 sections, 24 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exponential representations of intrinsic and dual volumes
  4. Applications to polynomial optimization
  5. Structural properties of intrinsic/dual volumes of sublevel sets
  6. Analogues of Löwner--John ellipsoids
  7. Nonnegative polynomial with sublevel sets of minimal intrinsic/dual volumes
  8. Computing intrinsic volumes via block--decomposition
  9. Arithmetic applications of the exponential representations
  10. A Lipschitz–type lattice discrepancy bound via intrinsic volumes
  11. Counting on rational subspaces and linear constraints
  12. Theta function asymptotics
  13. Appendix: On the choice of $\mathcal{P}_{n, d}$
  14. Acknowledgments.

Key Result

Lemma 3.1

For any $0 \leq j \leq n - 1$, let $f\in\mathcal{P}_{n,d}$ and let $E \in G(j, n)$ be a $j$--dimensional subspace of $\mathbb{R}^n$. Then, the following assertions hold true:

Theorems & Definitions (61)

  • Lemma 3.1: Projection/section operators
  • proof
  • Remark 3.2: Projection and section operators for quadratics
  • Lemma 3.3
  • Remark 3.4
  • proof : Proof of Lemma \ref{['lem.vol_j']}
  • Theorem 3.5: Laplace--Grassmannian representation
  • proof
  • Remark 3.6: Integrability
  • Remark 3.7
  • ...and 51 more