On the sum of powered distances to certain sets of points on the circle

Details On the sum of powered distances to certain sets of points on the circle On the sum of powered distances to certain sets of points on the circle

2011-08-23

arxiv.org/abs/1108.4652v1

Mathematics

Metric Geometry

to appear in Pacific J. Math

Scientific paper

In this paper we consider an extremal problem in geometry. Let $\lambda$ be a real number and $A$, $B$ and $C$ be arbitrary points on the unit circle $\Gamma$. We give full characterization of the extremal behavior of the function $f(M,\lambda)=MA^\lambda+MB^\lambda+MC^\lambda$, where $M$ is a point on the unit circle as well. We also investigate the extremal behavior of $\sum_{i=1}^nXP_i$, where $P_i, i=1,...,n$ are the vertices of a regular $n$-gon and $X$ is a point on $\Gamma$, concentric to the circle circumscribed around $P_1...P_n$. We use elementary analytic and purely geometric methods in the proof.

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