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Convex Functions

  • Rodney Coleman
Chapter
Part of the Universitext book series (UTX)

Abstract

Let X be a convex subset of a vector space V. We say that \(f : X\rightarrow \mathbb{R}\) is convex if for all x, y ε X and λ ε (0, 1) we have
$$f(\lambda x + (1 - \lambda )y) \leq \lambda f(x) + (1 - \lambda )f(y).$$
If the inequality is strict when xy, then we say that f is strictly convex. In this chapter we aim to look at some properties of these functions, in particular, when E is a normed vector space. For differentiable functions we will obtain a characterization, which will enable us to generalize the concept of a convex function.

Keywords

Vector Space Convex Function Convex Hull Convex Subset Differentiable Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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    Barvinok, A.: A Course in Convexity. American Mathematical Society, Providence, RI (2002)zbMATHGoogle Scholar
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    Borwein, J., Lewis, A.S.: Convex Analysis and Nonlinear Optimization, 2nd edn. Canadian Mathematical Society, Vancouver BC (2006)zbMATHCrossRefGoogle Scholar
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    Grünbaum, G., Ziegler, G.: Convex Polytopes, 2nd edn. Springer-Verlag, Berlin (2003)CrossRefGoogle Scholar
  4. 20.
    Rudin, W.: Functional Analysis. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  5. 22.
    Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Rodney Coleman
    • 1
  1. 1.Laboratoire Jean KuntzmannGrenobleFrance

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