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Convex cones and convex sets. (English) Zbl 0808.46015

Summary: Convex cones and convex sets are introduced without assuming they are vectorial, that is they are convex subcones and convex subsets of real vector spaces. The set of all nonvoid convex subsets of a convex set is a convex set, and the set of all nonvoid convex subsets of a convex cone is a convex cone. A convex cone is vectorial if and only if it satisfies the cancellation rule for convex cones. A convex set is vectorial if and only if it satisfies the cancellation rule for convex sets. Every convex set is a convex subset of some convex cone. Convex cones with constant multiplication are simple instances. Every convex cone whose power is strictly less than the power of the continuum has a constant multiplication. Every convex set whose power is strictly less than the power of the continuum has a unique convex cone structure with constant multiplication defining its convex set structure. There are equivalent conditions for a convex set to satisfy the cancellation rule for convex sets. Linear convex sets are easily characterized. For convex sets satisfying the one-dimensional injection rule, there is a smallest convex set failing to satisfy the cancellation rule for convex sets. Every convex cone is the image by a surjective convex cone map of a vectorial convex cone, and every convex set is the image by a surjective convex set map of a vectorial convex set. Associated with any convex cone, there is a largest vectorial convex cone. Associated with any convex set, there is a largest vectorial convex set.

MSC:

46A55 Convex sets in topological linear spaces; Choquet theory
52A05 Convex sets without dimension restrictions (aspects of convex geometry)
46A40 Ordered topological linear spaces, vector lattices
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