A note on average distances in digital sets

Abstract For any compact connected metric space S, there exists a unique nonnegative real number d such that, for any positive integer n and any points P1,…,Pn of S, there exists a point P in S whose average distance from the Pi's is exactly d . In this note we prove that for any finite connected digital set S and integer-valued metric defined on S, there exists a nonnegative integer d such that, for any positive integer n and any points P1,…,Pn of S, there exists a point in P in S whose average distance from the Pi's differs from d by at most 1 2 ; but D is not necessarily unique.