Diagonal Minkowski classes, zonoid equivalence, and stable laws

Molchanov, Ilya; Nagel, Felix (2020). Diagonal Minkowski classes, zonoid equivalence, and stable laws. Communications in contemporary mathematics, p. 1950091. World Scientific Publishing 10.1142/S0219199719500913

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We consider the family of convex bodies obtained from an origin symmetric convex body K by multiplication with diagonal matrices, by forming Minkowski sums of the transformed sets, and by taking limits in the Hausdorff metric. Support functions of these convex bodies arise by an integral transform of measures on the family of diagonal matrices, equivalently, on Euclidean space, which we call K-transform. In the special case, if K is a segment not lying on any coordinate hyperplane, one obtains the family of zonoids and the cosine transform. In this case two facts are known: the vector space generated by support functions of zonoids is dense in the family of support functions of origin symmetric convex bodies; and the cosine transform is injective. We show that these two properties are equivalent for general K. For K being a generalized zonoid, we determine conditions that ensure the injectivity of the K-transform. Relations to mixed volumes and to a geometric description of one-sided stable laws are discussed. The later probabilistic application motivates our study of a family of convex bodies obtained as limits of sums of diagonally scaled ℓp-balls.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science

UniBE Contributor:

Molchanov, Ilya and Nagel, Felix

Subjects:

500 Science > 510 Mathematics

ISSN:

1793-6683

Publisher:

World Scientific Publishing

Language:

English

Submitter:

Ilya Molchanov

Date Deposited:

04 May 2020 12:09

Last Modified:

10 May 2020 02:41

Publisher DOI:

10.1142/S0219199719500913

BORIS DOI:

10.7892/boris.138561

URI:

https://boris.unibe.ch/id/eprint/138561

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