Fundamentals of Convex Analysis pp 77-120 | Cite as

# Convex Cones in R^{n}

## 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

**x**

*∈*C, then λ

**x**

*∈*C for 0 ≤ λ

*∈*

**R**,

**x**

*∈*

**R**

^{ n }(see Figure 4.1.a for C in

**R**

^{2}). If we consider the set of points X = {

**x**}, then the

**cone generated by****X**is C = {y|y = λ

**x**, 0 ≤ λ

*∈*

**R**,

**x**

*∈*X}. And if

**0**

*∉*X and for each

**y(≠0)**

*∈*C there are unique

**x**

*∈*X with λ > 0 such that

**y**= λ

**x**, then X is termed a

*of C. Moreover, the point*

**base****0**

*∈*

**R**

^{ n }is termed the

*of a cone and is an element of every cone since*

**vertex****Y**=

**0x**=

**0**

*∈*C. 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## Preview

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