Annals of Functional Analysis

Convex components and multi-slices in real topological vector spaces

F. J. Garcia-Pacheco

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It is shown that, in a non-necessarily Hausdorff real topological vector space, if a subset is a countable disjoint union of convex sets closed in the subset, then those convex sets must be its convex components. On the other hand, by means of convex components we extend the notion of extreme point to non-convex sets, which entails a new equivalent reformulation of the Krein--Milman property (involving drops among other objects). Finally, we study the nature of convex functions and provide some results on their support in order to introduce the concept of multi-slice, that is, slices determined by convex functions (instead of by linear functions). Among other things, we prove that the boundary of a closed convex set with non-empty interior can be obtained as the set of support points of a certain lower semi-continuous convex function on that convex set.

Article information

Ann. Funct. Anal., Volume 6, Number 3 (2015), 73-86.

First available in Project Euclid: 17 April 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A03: Vector spaces, linear dependence, rank
Secondary: 46A55: Convex sets in topological linear spaces; Choquet theory [See also 52A07] 46B20: Geometry and structure of normed linear spaces

Krein--Milman property convex set convex component extreme point slice


Garcia-Pacheco, F. J. Convex components and multi-slices in real topological vector spaces. Ann. Funct. Anal. 6 (2015), no. 3, 73--86. doi:10.15352/afa/06-3-7.

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