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
$$|g(x) - g(y)|\; \leqslant \lambda ||x - y||{\text{ for all }}x,y \in \Theta$$


Open Subset Compact Subset Linear Space Differentiable Function Euclidean Norm 
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.


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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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