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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Unit lemniscates contained in the unit ball


Authors: Mau Hsiang Shih and Hann Tzong Wang
Journal: Proc. Amer. Math. Soc. 86 (1982), 451-454
MSC: Primary 30C10; Secondary 26C10, 30C15, 52A37
MathSciNet review: 671213
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Abstract: Let $ \{ {A_1},{A_2}, \ldots ,{A_\nu }\} \equiv A$ be a set of points in $ {E^n}$. Let $ E(A)$ be the set of points in $ {E^n}$ such that $ \Pi _{k = 1}^\nu \overline {p{A_k}} \leqslant 1$ (where $ \overline {p{A_k}} $ denotes the Euclidean distance between $ p$ and $ {A_k}$), and call this set the unit lemniscate with focal set $ A$. It is shown that if the vertices of a regular tetrahedron lie at the distance $ \delta \in (0,1)$ from the origin, then they are the foci of a unit lemniscate contained in the open unit ball of $ {E^3}$ if and only if the sign of $ 36 - 64\delta + 2{\delta ^2} - 64{\delta ^3} + 36{\delta ^4} + 27{\delta ^6}$ is positive.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0671213-8
PII: S 0002-9939(1982)0671213-8
Keywords: Complex polynomial, focal set, lemniscate, regular tetrahedron, sphere
Article copyright: © Copyright 1982 American Mathematical Society