On convex cones with infinitely many critical angles

This note deals with some cardinality issues concerning the set of critical angles of a convex cone K subset of R-d. Such set is referred to as the angular spectrum of the cone. In a recent work of ours, it has been shown that the angular spectrum of a polyhedral cone is necessarily finite and that its cardinality can grow at most polynomially with respect to the number of generators. In this note, we explore the case of nonpolyhedral cones. More specifically, we construct a cone whose angular spectrum is infinite (but possibly countable), and, what is harder to achieve, we construct a cone with noncountable angular spectrum. The construction procedure is highly technical in both cases, but the obtained results are useful for better understanding why some convex cones exhibit such a complicated angular structure.