Extending quasiconvex functions from uniformly convex sets
Abstract
Let $X$ be a normed space of a finite dimension at least two, and $C\subsetneq X$ a closed convex set with nonempty interior. We are interested in extending Lipschitz quasiconvex functions on $C$ to quasiconvex functions on $X$. We show that, unlike what holds for convex functions, in general one cannot obtain Lipschitz extensions (except for trivial cases). If we require just uniformly continuous or continuous extensions, such extendability properties for $C$ are shown to be characterized by some geometric properties of $C$.
Fields of Study & Subjects
Advanced Banach Space Theory
Optimization and Variational Analysis
Mathematical Inequalities and Applications
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 5 March 2026
- Language
- EN
- Document type
- Preprint