关于到圆周上某些点集的幂距离和
On the sum of powered distances to certain sets of points on the circle
摘要：
本文考虑几何中的一个极值问题。设$λ$为实数，$A$、$B$和$C$为单位圆$Γ$上的任意点。本文给出了单位圆上的点为$M$的函数$f(M,λ)=MA^λ+MB^λ+MC^λ$极值行为的充分刻画。我们还研究了$\sum_{i=1}^nXP_i$的极值行为，其中$P_i, i=1,...,n$是正则$n$ -边形的顶点，$X$是$Γ$上的一个点，与$P_1...P_n$外接的圆同心.我们使用初等分析和纯几何方法的证明。
英文摘要：
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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