Every closed convex set is the set of minimizers of some C1-smooth convex function
The authors show that for every closed convex set C in a separable Banach space there is a nonnegative C1 convex function f such that C = {x: f(x) = 0}. The key is to show this for a closed halfspace. This result has several attractive consequences. For example, it provides an easy proof that every...
| Autores: | , |
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| Tipo de documento: | artigo |
| Data de publicação: | 2002 |
| País: | España |
| Recursos: | Universidad Complutense de Madrid (UCM) |
| Repositório: | Docta Complutense |
| Idioma: | inglês |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/57021 |
| Acesso em linha: | https://hdl.handle.net/20.500.14352/57021 |
| Access Level: | Acceso aberto |
| Palavra-chave: | 517.98 Análisis funcional y teoría de operadores |
| Resumo: | The authors show that for every closed convex set C in a separable Banach space there is a nonnegative C1 convex function f such that C = {x: f(x) = 0}. The key is to show this for a closed halfspace. This result has several attractive consequences. For example, it provides an easy proof that every closed convex set is the Hausdorff limit of infinitely smooth convex bodies (Cn := {x: f(x) _ 1/n}) and that every continuous convex function is the Mosco limit of C1 convex functions. |
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