We can easily show that for every non-empty set $A$ in a normed space $X$,
$ \text{diam}(A) = 2\inf_{x \in X} \sup_{a \in A} \|a-x\|.$
This does not necessarily hold in the case of metric spaces. This variational characterization of diameter is used in a part of Alexei Kulik's book "Ergodic behavior of Markov processes" in the case of $X = \mathbb R$ equipped with the Euclidean norm. Are there any known name of this equality?