Cone domains separate FS-domains from RB-domains

2026/07/02 by Yuxu Chen · 1 voice
Mathematics · #math.GN

paper · pdf

Abstract

Let C be a closed, convex, pointed and generating cone in a finite-dimensional real vector space V, and let \( D_C=(-C)∪\\bot\\) be the negative cone with a new least element, ordered by the cone order. Keimel proved that these cone domains are FS-domains and asked whether they are always retracts of bifinite domains. We give a sharp answer: D_C is an RB-domain \Longleftrightarrow C is simplicial. Thus every non-simplicial proper cone gives an FS-domain which is not an RB-domain. The proof converts the RB approximation property into finite-valued C-isotone approximations of the identity. The analytic obstruction is elementary and finite-dimensional: first in Euclidean space, cone-upper sets are represented, up to null sets, as Lipschitz epigraphs; Rademacher's theorem, Fubini's theorem and integration by parts then force the matrix tested against any finite-valued isotone map to lie in the cone generated by the positive rank-one operators v⊗ℓ, v∈ C, ℓ∈ C^*. If such maps approximate the identity, the identity operator lies in this rank-one cone, which is possible exactly when the cone is simplicial. This answers Keimel's question in the negative for the Lorentz cone and other non-simplicial cones.

Citations

• Convex optimization

Discussions

• Yuxu Chen: Cone domains separate FS-domains from RB-domains https://arxiv.org/abs/2607.02251 https://arxiv.org/pdf/2607.02251 https://arxiv.org/html/2607.02251 [bsky, 0 points, 0 comments]

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