## Annals of Functional Analysis

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

### Convex components and multi-slices in real topological vector spaces

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 17 April 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.afa/1429286033

**Digital Object Identifier**

doi:10.15352/afa/06-3-7

**Mathematical Reviews number (MathSciNet)**

MR3336906

**Zentralblatt MATH identifier**

06441309

**Subjects**

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.afa/1429286033