On convex cones with infinitely many critical angles

Abstract

This note deals with some cardinality issues concerning the set of critical angles of a convex cone . 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.

Keywords:

• Convex cones
• Critical angles
• Angular spectra
• Cantor ternary set

Keywords:

• Mathematics Subject Classifications 2000:
• 52A40
• 90C26

Notes

¹ As pointed out by one of the referees, the criticality conditions stated in (2) can also be derived by using a direct geometric argument. Assuming u≠ −v, denote by P the (d−2)-dimensional subspace orthogonal to the vectors u,v and by Γ₁,Γ₂ the hyperplanes containing P ∪ {u} and P∪{v}, respectively. The fact that u and v are unit vectors in K achieving the maximal angle (1) implies that Γ₁ and Γ₂ are support hyperplanes of K. Observe now that the vectors v−⟨ u,v ⟩ u and u−⟨ u,v⟩ v are orthogonal to Γ₁ and Γ₂, respectively.