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Question: we say a set c of points in rn is...

Question details

We say a set C of points in Rn

is convex if for every pair x, y ∈ 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 ∈ C and any
λ ∈ (0, 1) we have λx + (1 − λ)y ∈ C. Note that λx + (1 − λ)y = y + λ(x − y) which for
λ ∈ [0, 1], yields the line segment joining x and y. Let A be an m × n matrix and b a given
vector in Rm. Show that

C = {x ∈ Rn

: Ax ≤ b, x ≥ 0}
is a convex set. (That is, the domain of a LP is convex).

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