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.
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Notes
- 1.
Although X is a normed space throughout this chapter, it is clear that these basic definitions require only that X be a vector space.
- 2.
We say that f is positively homogeneous if f(tx)= tf(x) whenever t is a positive scalar. Subadditivity is the property f(x+y) ⩽ f(x)+f(y).
- 3.
Each x∈ K 1 admits r(x)>0 such that B(x,2r(x))⊂ X∖ K 2. Let { B(x i ,r(x i ))} be a finite subcovering of K 1. Then we may take \(V\,=\,\cap_{\:i}\,\,B^{\,\displaystyle{\circ}} ( 0 , r(x_{\, i})) \).
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© 2013 Springer-Verlag London
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Clarke, F. (2013). Convex sets and functions. In: Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics, vol 264. Springer, London. https://doi.org/10.1007/978-1-4471-4820-3_2
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DOI: https://doi.org/10.1007/978-1-4471-4820-3_2
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Publisher Name: Springer, London
Print ISBN: 978-1-4471-4819-7
Online ISBN: 978-1-4471-4820-3
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