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Problem 4. Which of the following sets are convex? (You need to justify your answer with clear logical reasoning.) (a) The set of points closer to a given point than a given set, i.e., {x| ||X-X0||2 ≤ ||X-y||2 Vy € S}, (b) (c) where SC Rn. The set of points closer to one set than another, i.e., {x dist(x, S) < dist(x, T)}, where S, TCR, and The set dist(x, S) = inf{||x - z||2| Z = S}. S = {x| (x + y) = S₁, Vy S₂}, where S₁, S₂ CR" with S₁ convex. (d) The set of points whose distance to a point a does not exceed a fixed fraction 0 ≤0 ≤ 1 of the distance to another point b, i.e., the set {x| ||xa||2 ≤0||x - b||2}.

Problem 4. Which of the following sets are convex? (You need to justify your answer with clear logical reasoning.) (a) The set of points closer to a given point than a given set, i.e., {x| ||X-X0||2 ≤ ||X-y||2 Vy € S}, (b) (c) where SC Rn. The set of points closer to one set than another, i.e., {x dist(x, S) < dist(x, T)}, where S, TCR, and The set dist(x, S) = inf{||x - z||2| Z = S}. S = {x| (x + y) = S₁, Vy S₂}, where S₁, S₂ CR" with S₁ convex. (d) The set of points whose distance to a point a does not exceed a fixed fraction 0 ≤0 ≤ 1 of the distance to another point b, i.e., the set {x| ||xa||2 ≤0||x - b||2}.

Advanced Engineering Mathematics
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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Problem 4. Which of the following sets are convex? (You need to justify your answer with clear logical reasoning.) (a) The set of points closer to a given point than a given set, i.e., {x| ||X-X0||2||x-y||2 Vy € S}, (b) (c) where SCR". The set of points closer to one set than another, i.e., {x dist(x, S) < dist(x, T)}, where S, TCR, and The set dist(x, S) inf{||xz||2| Z = S}. S = {x| (x + y) E S₁, Vy S₂}, where S₁, S₂R" with S₁ convex. (d) The set of points whose distance to a point a does not exceed a fixed fraction 0 ≤0 ≤ 1 of the distance to another point b, i.e., the set {x| ||xa||2 ≤0|x - b||2}.
Transcribed Image Text:Problem 4. Which of the following sets are convex? (You need to justify your answer with clear logical reasoning.) (a) The set of points closer to a given point than a given set, i.e., {x| ||X-X0||2||x-y||2 Vy € S}, (b) (c) where SCR". The set of points closer to one set than another, i.e., {x dist(x, S) < dist(x, T)}, where S, TCR, and The set dist(x, S) inf{||xz||2| Z = S}. S = {x| (x + y) E S₁, Vy S₂}, where S₁, S₂R" with S₁ convex. (d) The set of points whose distance to a point a does not exceed a fixed fraction 0 ≤0 ≤ 1 of the distance to another point b, i.e., the set {x| ||xa||2 ≤0|x - b||2}.
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