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

 

A problem on a geometric property of lemniscates


Author: J. S. Hwang
Journal: Proc. Amer. Math. Soc. 82 (1981), 390-392
MSC: Primary 26C10; Secondary 30C15
MathSciNet review: 612726
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Abstract: Let $ {R^3}$ be the Eulcidean space and let $ {p_n}$ be the product defined by $ {p_n}(W,{W_k}) = \Pi _{k = 1}^n\left\vert {W - {W_k}} \right\vert$, $ W$, $ {W_k} \in {R^3}$, where $ \left\vert {W - {W_k}} \right\vert$ is the distance between $ W$ and $ {W_k}$. Let $ C(n)$ be the class of all such products with the same degree $ n$. For any product $ p$, we call $ E(p) = \{ W:p(W) \leqslant 1\} $ the lemniscate of $ p$. We recently proved that if $ {p_n}(W,{W_k})$ and $ p_n^*(W,W_k^*)$ are two products in $ C(n)$ such that $ E({p_n}) \subseteq E(p_n^*)$, and if all zeros $ {W_k}$ of $ {p_n}$ lie on the same plane, then $ {p_n} \equiv p_n^*$. We then asked whether this result is sharp. In this note, we answer this question in the affirmative sense.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0612726-3
PII: S 0002-9939(1981)0612726-3
Keywords: Product, lemniscate, extremal property
Article copyright: © Copyright 1981 American Mathematical Society