Constructive Minimal Cuscos

Borwein, Jonathan M. and Kortezov, Ivaylo (2004) Constructive Minimal Cuscos. C.R. Bulg. Acad. Sci., 57 (12). pp. 9-12.

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    A (set-valued) map F : Z → X is called an usco, if it is upper semicontinuous and F(z) is a nonempty compact set for every z ∈ Z. An usco map is called a minimal usco if it is minimal with respect to the graph inclusion among all usco maps with the same domain. The minimal usco maps arguably form the most important subclass of usco maps and appears in a wide variety of areas (see e.g. [2, 5]). Additionally, muscos represent a very good multivalued analogue of continuous mappings. Let us denote by ‘UCMU’ the statement “each usco contains a minimal usco” (here “contains” refers to graph inclusion). Now UCMU is easy to prove via the Zorn’s lemma. This is sometimes considered as a drawback, since it makes it impossible for the reader to “touch” the obtained map; in addition, the weirdness of some of its consequences can even make it troublesome to “believe” in minimal uscos. We can easily see that in general this drawback cannot be bypassed; in Proposition 1 below we show that the full Axiom of Choice (AC) can be derived from UCMU.

    Item Type: Article
    Additional Information: pubdom TRUE
    Subjects: 49-xx Calculus of variations and optimal control; optimization
    46-xx Functional analysis
    Faculty: UNSPECIFIED
    Depositing User: Users 1 not found.
    Date Deposited: 08 Oct 2004
    Last Modified: 12 Jan 2015 15:08

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