Center of Distances and Central Cantor Sets

We study a recently discovered metric invariant - the center of distances. The center of distances of a nonempty subset A of a metric space (X, d) is defined by S(A) := {alpha is an element of [0, + infinity) : for all x is an element of A there exists y is an element of A d(x, y) = alpha}. Given a nonincreasing sequence (a(n)) of positive numbers converging to 0, the set E(a(n)) := {x is an element of R: there exists A subset of N x = Sigma(n is an element of A) a(n)} is called the achievement set of the sequence (a(n)). This new invariant is particularly useful in investigating achievability of sets on the real line. We concentrate on computing the centers of distances of central Cantor sets. Any central Cantor set C is an achievement set of exactly one fast convergent series Sigma a(n), and consequently S(C) superset of {0} boolean OR {a(n) : n is an element of N). We try to check which central Cantor sets have the minimal possible center of distances and which have not.