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

TL;DR

This paper characterizes the extremal values of sums of powered distances between points on the circle, providing a complete geometric and analytic analysis for specific configurations involving points on the circle.

Contribution

It offers a full characterization of extremal behavior of distance sums involving points on the circle, extending to regular polygons and using elementary methods.

Findings

Explicit extremal configurations identified for the sum of powered distances.

Complete geometric and analytic characterization of the extremal behavior.

Results applicable to points on the circle and regular polygons.

Abstract

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.