Convex Functions

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


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.


Vector Space Convex Function Convex Hull Convex Subset Differentiable Function 
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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

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

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