Non-zero boundaries of Leibniz half-spaces
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Abstract:
It is proved that for any $d\geq 3$, there exists a norm $\|\cdot \|$ and two points $a$, $b$ in $\mathbb {R}^d$ such that the boundary of the Leibniz half-space $H(a,b)=\{x\in \mathbb {R}^d: \|x-a\|\leq \|x-b\|\}$ has non-zero Lebesgue measure. When $d=2$, it is known that the boundary must have zero Lebesgue measure.References
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Additional Information
- Fuchang Gao
- Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83843
- MR Author ID: 290983
- 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
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 1757-1762
- MSC (2000): Primary 46B20
- DOI: https://doi.org/10.1090/S0002-9939-04-07732-9
- MathSciNet review: 2120275