Average distances in compact connected spaces

摘要: We give a simple proof of the fact that compact, connected topological spaces have “average distance property”. For metric space (X, d), this asserts existence unique number = a(X) such that, given finitely many points x1, …, xn ∈ X, then there is some y X withWe examine possible values , for subsets finite dimensional normed spaces. example, if diam(X) denotes diameter convex set in euclidean space, ≤ diam(X)/√2 . On other hand, a(X)/diam(X) can be arbitrarily close to 1 non-convex sets sufficiently large dimension.