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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-xo||2|x-y|2 Vye S}, (b) where SC Rn. The set of points closer to one set than another, i.e., {x dist(x, S) < dist(x, T')},

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-xo||2|x-y|2 Vye S}, (b) where SC Rn. The set of points closer to one set than another, i.e., {x dist(x, S) < dist(x, T')},

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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Which sets are convex

please do with logical reasoning and if possible with mathematical analysis

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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