Advances in Operator Theory

Lipschitz properties of convex functions

Stefan Cobzaş

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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

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

First available in Project Euclid: 4 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46N10: Applications in optimization, convex analysis, mathematical programming, economics
Secondary: 26A16: Lipschitz (Hölder) classes 26A51: Convexity, generalizations 46A08: Barrelled spaces, bornological spaces 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) 46A40: Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42] 46B40: Ordered normed spaces [See also 46A40, 46B42]

convex function convex operator Lipschitz property ordered locally convex space cone normal cone normed lattice barrelled space metrizale locally convex space metric linear space


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

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