# Lipschitz properties of convex functions

Document Type: Original Article

Author

Babes-Bolyai University, Department of Mathematics

Abstract

The present paper is concerned with Lipschitz properties of convex mappings. One considers the general context of mappings defined on an open convex subset $Omega$ of a locally convex space $X$ and taking values in a locally convex space $Y$ ordered by a normal cone.
One proves also equi-Lipschitz properties for pointwise bounded families of continuous convex
mappings, provided the source space $X$ is barrelled.
Some results on Lipschitz properties of continuous convex functions defined on metrizable topological vector spaces are included as well.
The paper has a methodological character - its aim is to show that some geometric properties (monotonicity of the slope, the normality of the seminorms) allow to extend the proofs from the scalar case to the vector one. In this way the proofs become more transparent and natural.

Keywords

### Histroty

• Receive Date: 03 October 2016
• Revise Date: 31 December 2016
• Accept Date: 01 January 2017