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The zonoid algebra, generalized mixed volumes, and random determinants

Paul Breiding, Peter Bürgisser, Antonio Lerario, Léo Mathis

TL;DR

This work develops a comprehensive framework that unifies zonoids, mixed volumes, and random determinants by constructing the zonoid algebra: a graded, commutative, associative, partially ordered ring built from exterior powers of a Euclidean space using a tensor (and wedge) product on zonoids. It shows that every Euclidean multilinear map induces a corresponding multilinear map on zonoids, and introduces the length functional, which recovers the first intrinsic volume and expresses mixed volumes as MV(K_1,...,K_m)= (1/m!) ell(K_1∧…∧K_m). The paper further extends the theory to the complex setting via the mixed $J$-volume, proving Vitale-type formulas for the expected determinant of random matrices with complex blocks, and studies the extension of these notions to polytopes (with a Kazarnovskii-type variant) while identifying obstruction to full extension to all convex bodies. Overall, the framework provides a probabilistic-geometric bridge with potential impacts on probabilistic intersection theory and a Schubert-calculus-like view in random position geometry.

Abstract

We show that every multilinear map between Euclidean spaces induces a unique, continuous, Minkowski multilinear map of the corresponding real cones of zonoids. Applied to the wedge product of the exterior algebra of a Euclidean space, this yields a multiplication of zonoids, defining the structure of a commutative, associative, and partially ordered ring, which we call the zonoid algebra. This framework gives a new perspective on classical objects in convex geometry, and it allows to introduce new functionals on zonoids, in particular generalizing the notion of mixed volume. We also analyze a similar construction based on the complex wedge product, which leads to the new notion of mixed $J$-volume. These ideas connect to the theory of random determinants.

The zonoid algebra, generalized mixed volumes, and random determinants

TL;DR

This work develops a comprehensive framework that unifies zonoids, mixed volumes, and random determinants by constructing the zonoid algebra: a graded, commutative, associative, partially ordered ring built from exterior powers of a Euclidean space using a tensor (and wedge) product on zonoids. It shows that every Euclidean multilinear map induces a corresponding multilinear map on zonoids, and introduces the length functional, which recovers the first intrinsic volume and expresses mixed volumes as MV(K_1,...,K_m)= (1/m!) ell(K_1∧…∧K_m). The paper further extends the theory to the complex setting via the mixed -volume, proving Vitale-type formulas for the expected determinant of random matrices with complex blocks, and studies the extension of these notions to polytopes (with a Kazarnovskii-type variant) while identifying obstruction to full extension to all convex bodies. Overall, the framework provides a probabilistic-geometric bridge with potential impacts on probabilistic intersection theory and a Schubert-calculus-like view in random position geometry.

Abstract

We show that every multilinear map between Euclidean spaces induces a unique, continuous, Minkowski multilinear map of the corresponding real cones of zonoids. Applied to the wedge product of the exterior algebra of a Euclidean space, this yields a multiplication of zonoids, defining the structure of a commutative, associative, and partially ordered ring, which we call the zonoid algebra. This framework gives a new perspective on classical objects in convex geometry, and it allows to introduce new functionals on zonoids, in particular generalizing the notion of mixed volume. We also analyze a similar construction based on the complex wedge product, which leads to the new notion of mixed -volume. These ideas connect to the theory of random determinants.

Paper Structure

This paper contains 20 sections, 51 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Zonoids
  3. Convex bodies and support functions
  4. Zonoids and random convex bodies
  5. Virtual zonoids
  6. Virtual zonoids and measures
  7. Tensor product of zonoids
  8. Continuity of the tensor product
  9. Multilinear maps induced on zonoids
  10. Zonoids and multilinear maps
  11. Zonoid algebra
  12. Grassmannian zonoids
  13. Hodge duality
  14. Mixed volumes and random determinants
  15. Length functional and intrinsic volumes
  16. ...and 5 more sections

Key Result

Proposition 2.1

The map $h\colon (\mathcal{K}(V), d_H) \to (C(S(V)), \Vert \cdot\Vert_{\infty}),\, K\mapsto \bar{h}_K$ is a Minkowski linear isometric embedding. That is, for $K,L\in \mathcal{K}(V)$ and $\lambda \geq 0$, In particular, $\|K\| = \|\bar{h}_K\|_{\infty}$. Moreover,

Theorems & Definitions (137)

  • Proposition 2.1
  • proof
  • Definition 2.2: Zonoids
  • Definition 2.3: Zonoid associated to random variable
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 127 more