Robust test problems between two approximately known simple hypotheses can be formalized as minimax test problems between two composite hypotheses. We show that if the composite hypotheses can be described in terms of alternating capacities of order 2 (in the sense of Choquet), then the minimax tests are ordinary Neyman-Pearson tests between a fixed representative pair of simple hypotheses; moreover, the condition is in a certain sense also necessary. All the neighborhoods customarily used to formalized approximate knowledge happen to have this particular structure.
"Minimax Tests and the Neyman-Pearson Lemma for Capacities." Ann. Statist. 1 (2) 251 - 263, March, 1973. https://doi.org/10.1214/aos/1176342363