Non-zero boundaries of Leibniz half-spaces

Gao, Fuchang

doi:10.1090/S0002-9939-04-07732-9

Non-zero boundaries of Leibniz half-spaces
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Proc. Amer. Math. Soc. 133 (2005), 1757-1762 Request permission

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.

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