Uncertainty of measuring complex-valued physical quantities can be described by complex sets. These sets can have complicated shapes, so we would like to find a good approximating family of sets. Which approximating family is the best? We reduce the corresponding optimization problem to a geometric one: namely, we prove that, under some reasonable conditions, an optimal family must be shift-, rotation- and scale-invariant. We then use this geometric reduction to conclude that the best approximating low-dimensional families consist of sets with linear or circular boundaries. This result is consistent with the fact that such sets have indeed been successful in computations. It stimulates to study further candidates.