Azagra Rueda, DanielFerrera Cuesta, Juan2023-06-202023-06-2020021088-682610.1090/S0002-9939-02-06695-9https://hdl.handle.net/20.500.14352/57021The 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.engjournal articlehttp://www.ams.org/proc/open access517.98Análisis funcional y teoría de operadores