The average distance property for subsets of Euclidean space (Q1092445)

It is known that for any compact connected metric space X, there is a unique real number a(X) such that, for any finite collection \(x_ 1,x_ 2,...,x_ m\) of points in X, there is a \(y\in X\) with \(\frac{1}{m}\sum ^{m}_{i=1}d(x_ i,y)=a(X)\). In some sense, a(X) measures the average distance between elements of X. Clearly a(X)\(\leq diam(X)\). We establish fairly sharp bounds for \(k_ n=\max \{a(X)/diam(X):\) \(X\subset {\mathbb{R}}^ n\}\). In particular \(k_ 2<0.7183\).