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

In this paper we consider an extremal problem in geometry. Let $λ$ be a real number and $A$, $B$ and $C$ be arbitrary points on the unit circle $Γ$. We give full characterization of the extremal behavior of the function $f(M,λ)=MA^λ+MB^λ+MC^λ$, 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 $Γ$, 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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