### Lipschitz properties of convex functions

Stefan Cobzaş

#### 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 convexmappings, 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.

#### Article information

Source
Adv. Oper. Theory, Volume 2, Number 1 (2017), 21-49.

Dates
First available in Project Euclid: 4 December 2017

https://projecteuclid.org/euclid.aot/1512431512

Digital Object Identifier
doi:10.22034/aot.1610.1022

Mathematical Reviews number (MathSciNet)
MR3730353

Zentralblatt MATH identifier
1379.46056

#### Citation

Cobzaş, Stefan. Lipschitz properties of convex functions. Adv. Oper. Theory 2 (2017), no. 1, 21--49. doi:10.22034/aot.1610.1022. https://projecteuclid.org/euclid.aot/1512431512

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