Concave Functions

  • Gerhard Winkler
Part of the Applications of Mathematics book series (SMAP, volume 27)


A subset C of a linear space E is called convex if for all x, yC the line-segment
$$[x,y] = \{ \lambda x + (1 - \lambda )y:0 \leqslant \lambda \leqslant 1\}$$
is contained in C. For x(1),…,x(n)E and λ(1),…, λ(n) ≥ 0, ∑ i λ(i) = 1, the element x = ∑ i λ(i)x(i) is called a convex combination of the elements x(i). For x = (x1,…,x d ), y = (x1,…,y d ) ∈ R d , the symbol 〈x, y〉 denotes the Euclidean scalar product ∑ i x i y i and ‖ · ‖ denote Euclidean norm. A real-valued function g on a subset Θ of R d is called Lipschitz continuous if there is λ > 0 such that
$$|g(x) - g(y)|\; \leqslant \lambda ||x - y||{\text{ for all }}x,y \in \Theta$$




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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Gerhard Winkler
    • 1
  1. 1.Mathematical InstituteLudwig-Maximilians UniversitätMünchenGermany

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