The zonoid algebra, generalized mixed volumes, and random determinants
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866914635433639936 |
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| author | Breiding, Paul Bürgisser, Peter Lerario, Antonio Mathis, Léo |
| author_facet | Breiding, Paul Bürgisser, Peter Lerario, Antonio Mathis, Léo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_14996 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The zonoid algebra, generalized mixed volumes, and random determinants Breiding, Paul Bürgisser, Peter Lerario, Antonio Mathis, Léo Metric Geometry 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. |
| title | The zonoid algebra, generalized mixed volumes, and random determinants |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2109.14996 |