Unit lemniscates contained in the unit ball
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- by Mau Hsiang Shih and Hann Tzong Wang
- Proc. Amer. Math. Soc. 86 (1982), 451-454
- DOI: https://doi.org/10.1090/S0002-9939-1982-0671213-8
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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.References
- P. Erdös, Advanced problems and solutions 4229, Amer. Math. Monthly 55 (1948), 171.
- P. Erdős and J. S. Hwang, On a geometric property of lemniscates, Aequationes Math. 17 (1978), no. 2-3, 344–347. MR 500360, DOI 10.1007/BF01818573
- J. S. Hwang, A problem on a geometric property of lemniscates, Proc. Amer. Math. Soc. 82 (1981), no. 3, 390–392. MR 612726, DOI 10.1090/S0002-9939-1981-0612726-3 H. T. Wang, On a problem of Erdös and Hwang, Technical report, Chung-Yuan University, 1980.
Bibliographic Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 86 (1982), 451-454
- MSC: Primary 30C10; Secondary 26C10, 30C15, 52A37
- DOI: https://doi.org/10.1090/S0002-9939-1982-0671213-8
- MathSciNet review: 671213