Nota de FísicaPublicado
Convex Cones and Convex Sets
Leopoldo Nachbin
1992
Dados da publicação
- Autores
- Leopoldo Nachbin
- Tipo
- Nota de Física
- Status
- Publicado
- Data de publicação
- 1992
- Descrição
- Convex cones (Definitions 2 and 3) and convex sets (Definition 20) 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 (Definition 24). A convex cone is vectorial if and only if it satisfies the cancellation rule for convex cones (Proposition 10). A convex set is vectorial if and only if it satisfies the cancellation rule for convex sets (Proposition 34). Every convex set is a convex subset of some convex cone (Proposition 31). Convex cones with constant multiplication are simple instances (Proposition 18). Every convex cone whose power is strictly less than the power of the continuum has a constant multiplication (Proposition 40). 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 (Proposition 41). There are equivalent conditions for a convex set to satisfy the cancellation rule for convex sets (Proposition 44). Linear convex sets are easily characterized (Proposition 46). For convex sets satisfying the one dimensional injection rule, there is a smallest convex set failing to satisfy the cancellation rule for convex sets (Proposition 48). 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 (Proposition 35). Associated with any convex cone, there is a largest vectorial convex cone (Propositions 13 and 10). Associated with any convex set, there is a largest vectorial convex set (Proposition 42).
- Ano
- 1992
- Número
- CBPF-NF-001/92
- Vínculo com tese
- Não
- Vínculo com bolsa
- Não