We say a set C of points in Rn is convex if for every pair x

We say a set C of points in R^n is convex if for every pair x, y elementof C, all points on the line segment joining x and y are in C. Thus C is a convex set if for every pair x. y elementof C and any lambda elementof (0, 1). then lambda x + (1 - lambda)y elementof C. Let A be an m times n matrix and b a given vector in R^m. Show that F = {x elementof R^n: Ax lessthanorequalto b, x greaterthanorequalto 0} is a convex set. When you begin to think of this problem you should understand that lambda a + (1 - lambda)b for lambda elementof [0, 1] is a weighted average of a and b and the set {lambda u + (1 - lambda)v: lambda elementof [0, 1]} = {v + lambda (u - v):lambda elementof [0, 1]} is the line segment joining u and v.

Solution

Solution :

Let x₁ and x₂ be in the set. Then   Ax₁ b       and   Ax₂ b

Consider ₁(>0) and   ₂ (>0) such that ₁+₂= 1

Then,

   ₁(Ax₁) +₂(Ax₂) b(₁+₂)

₁(Ax₁) +₂(Ax₂) b

             A( ₁x₁ + ₁x₂) b Thus, { x :Axb } is convex.