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Erschienen in:
2013 | OriginalPaper | Buchkapitel
2. Convex sets and functions
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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Abstract
The class of convex sets plays a central role in the theory. The subset C of the normed space X is said to be convex if The separation of convex sets is the issue treated by the celebrated Hahn-Banach theorem, often said to be the most basic tool of classical functional analysis. We also study convex functions, a more modern consideration. Such functions, which turn out to have special regularity properties, will play a big role in things to come. We also introduce lower semicontinuous and extended-valued functions, which are important later in optimization. Support functions and indicator functions are important examples of these.
$$t\in\, (0 , 1)\,,\:\:\: x , y\: \in \:C\:\: \Longrightarrow\:\: (1-t) x+ t y\: \in \:C\,.$$