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Is this property related to the q-extent of a metric space X which is the maximum average distance between q points in X? i.e. $xt_q(X):=\max_{x_1,\dots,x_q}xt_q(x_1 ,\dots ,x_q)$ where $xt_q(x_1 ,\dots,x_q) = {q\choose 2}^{-1}\sum_{i<j} \operatorname{dist}(x_i , x_j) .$