Finite-stage compactness for nonpolyhedral exponent cones in dimensions at least three

Determine whether, in dimensions n≥3 and for a compact convex set S⊂R^n_+ whose cone R_+S is not rational polyhedral, every compact S-convex set satisfies the equivalent finite-stage conditions of Theorem B, and consequently whether the Oka–Weil approximation property holds for every such compact S-convex set.

Background

The paper studies Oka–Weil approximation by the sparse polynomial ring consisting of polynomials whose exponent vectors lie in the cone R_+S generated by a compact convex set S⊂R^n_+ containing the origin. For a compact S-convex set K, Theorem B establishes equivalence between the Oka–Weil property, naturality of the uniform algebra spectrum, a corona-type certificate, containment in a bounded sparse polyhedron, and compactness of the hull with respect to a finitely generated monomial subring.

The paper proves these conditions for rational polyhedral exponent cones, for every S in dimension two, for compact sets contained in the torus, for several nonpolyhedral classes in arbitrary dimensions, and for compact S-convex sets lying in coordinate subspaces of dimension at most two. The unresolved issue is whether the same conclusions hold universally in dimensions n≥3 when the exponent cone is nonpolyhedral. Remark 2.14 reformulates the issue as asking whether the uniform algebra spectrum can contain characters other than point evaluations on K.