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Convex Cones in Rn

  • Michael J. Panik
Chapter
Part of the Theory and Decision Library book series (TDLB, volume 24)

Abstract

A cone C in R n is a set of points such that if x C, then so is every nonnegative scalar multiple of x, i.e., if xC, then λxC for 0 ≤ λR, xR n (see Figure 4.1.a for C in R2). If we consider the set of points X = {x}, then the cone generated by X is C = {y|y = λx, 0 ≤ λ R, xX}. And if 0 X and for each y(≠0)C there are unique xX with λ > 0 such that y = λx, then X is termed a base of C. Moreover, the point 0R n is termed the vertex of a cone and is an element of every cone since Y = 0x = 0C. In general, C is “not a subspace” of R n since the definition of C holds only for λ ≥ 0 and not all λ.

Keywords

Convex Cone Independent Vector Extreme Solution Supporting Hyperplane Homogeneous Linear System 
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 Science+Business Media Dordrecht 1993

Authors and Affiliations

  • Michael J. Panik
    • 1
  1. 1.Department of EconomicsUniversity of HartfordWest HartfordUSA

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