Global optimality under amenable symmetry constraints

Consider a convex function that is invariant under an group of transformations. If it has a minimizer, does it also have an invariant minimizer? Variants of this problem appear in nonparametric statistics and in a number of adjacent fields. The answer depends on the choice of function, and on what one may loosely call the geometry of the problem -- the interplay between convexity, the group, and the underlying vector space, which is typically infinite-dimensional. We observe that this geometry is completely encoded in the smallest closed convex invariant subsets of the space, and proceed to study these sets, for groups that are amenable but not necessarily compact. We then apply this toolkit to the invariant optimality problem. It yields new results on invariant kernel mean embeddings and risk-optimal invariant couplings, and clarifies relations between seemingly distinct ideas, such as the summation trick used in machine learning to construct equivariant neural networks and the classic Hunt-Stein theorem of statistics.