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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Metric inequalities and the zonoid problem


Author: H. S. Witsenhausen
Journal: Proc. Amer. Math. Soc. 40 (1973), 517-520
MSC: Primary 52A40
DOI: https://doi.org/10.1090/S0002-9939-1973-0390916-0
MathSciNet review: 0390916
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Abstract: For normed spaces the hypermetric and quasihypermetric properties are equivalent and imply the quadrilateral property. The unit ball of a Minkowski space is a zonoid if and only if the dual space is hypermetric. The unit ball of $ l_p^n$ is not a zonoid for $ n = 3,p < \log 3/\log 2$, and for $ p \leqq 2 - {(2n\log 2)^{ - 1}} + o({n^{ - 1}})$. The elliptic spaces $ {\mathcal{E}^d},d > 1$, are not quasihypermetric.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0390916-0
Keywords: Metric inequalities, zonoids, hypermetric spaces, elliptic geometry
Article copyright: © Copyright 1973 American Mathematical Society