MathDB

Problem 6

Part of 2013 CIIM

Problems(1)

CIIM 2013 Problem 6

Source:

8/9/2016
Let (X,d)(X,d) be a metric space with d:X×X→R≥0d:X\times X \to \mathbb{R}_{\geq 0}. Suppose that XX is connected and compact. Prove that there exists an α∈R≥0\alpha \in \mathbb{R}_{\geq 0} with the following property: for any integer n>0n > 0 and any x1,…,xn∈Xx_1,\dots,x_n \in X, there exists x∈Xx\in X such that the average of the distances from x1,…,xnx_1,\dots,x_n to xx is α\alpha i.e. d(x,x1)+d(x,x2)+⋯+d(x,xn)n=α.\frac{d(x,x_1)+d(x,x_2)+\cdots+d(x,x_n)}{n} = \alpha.
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