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Proceedings of the American Mathematical Society

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

 

 

Non-zero boundaries of Leibniz half-spaces


Author: Fuchang Gao
Journal: Proc. Amer. Math. Soc. 133 (2005), 1757-1762
MSC (2000): Primary 46B20
DOI: https://doi.org/10.1090/S0002-9939-04-07732-9
Published electronically: November 19, 2004
MathSciNet review: 2120275
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Abstract: It is proved that for any $d\geq 3$, there exists a norm $\Vert\cdot\Vert$ and two points $a$, $b$ in $\mathbb{R}^d$ such that the boundary of the Leibniz half-space $H(a,b)=\{x\in \mathbb{R}^d: \Vert x-a\Vert\leq\Vert x-b\Vert\}$ has non-zero Lebesgue measure. When $d=2$, it is known that the boundary must have zero Lebesgue measure.


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Additional Information

Fuchang Gao
Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83843

DOI: https://doi.org/10.1090/S0002-9939-04-07732-9
Keywords: Leibniz half-space, Voronoi region, metric entropy
Received by editor(s): August 26, 2003
Received by editor(s) in revised form: February 17, 2004
Published electronically: November 19, 2004
Additional Notes: This research was partially supported by NSF grant EPS-0132626 and a seed grant from the University of Idaho.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.