Determine with proof if each of the following sets below is a convex set? 1 (a) {x en: x ≤r}, r> 0 is a real number. (b) {x en: Ax=b}, A is an m x n matrix and bem. (c) {x €": x ≥ 0}, where x ≥ 0 means that every component of x is nonnegative. (d) A linear variety. (e) If S₁ and S₂ are convex sets, then S₁ + S₂ = {x: X = V₁ + V2, V₁ € S₁, V2 € S₂} is also a convex set. (f) The intersection of any collection of convex sets is convex. To prove that a set is convex, we have to prove that for any x, y € , and any 0 ≤ a ≤ 1, the point ax + (1 - a)y is also in .
Determine with proof if each of the following sets below is a convex set? 1 (a) {x en: x ≤r}, r> 0 is a real number. (b) {x en: Ax=b}, A is an m x n matrix and bem. (c) {x €": x ≥ 0}, where x ≥ 0 means that every component of x is nonnegative. (d) A linear variety. (e) If S₁ and S₂ are convex sets, then S₁ + S₂ = {x: X = V₁ + V2, V₁ € S₁, V2 € S₂} is also a convex set. (f) The intersection of any collection of convex sets is convex. To prove that a set is convex, we have to prove that for any x, y € , and any 0 ≤ a ≤ 1, the point ax + (1 - a)y is also in .
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Determine with proof if each of the following sets below is a convex set?
1
(a) {x €¹: x ≤ r}, r> 0 is a real number.
(b) {x €¹: Ax = b}, A is an m × n matrix and bem.
(c) {x €¹: x ≥ 0}, where x ≥ 0 means that every component of x is
nonnegative.
(d) A linear variety.
(e) If S₁ and S₂ are convex sets, then
S₁ + S₂
{x: x
is also a convex set.
= V₁ + V₂, V₁ € S1, V₂ € S₂}
(f) The intersection of any collection of convex sets is convex.
To prove that a set is convex, we have to prove that for any x, y € ,
and any 0 ≤ a ≤ 1, the point ax + (1 - a)y is also in .
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