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3. Use the definition of convex sets to answer the following: Show that if the sets S and T are convex, then SOT is convex. Show that the intersection of any number of convex sets is convex. A hyperplane in Rd is a set of points of the form {r: ax=b} where a € Rd and b E R. Show that hyperplanes are convex. Hint: If you're having trouble seeing why this is true in general, try the problem with a simple concrete example in 2-dimensions. La (d) A halfspace Rd is a set of points of the form {x: a¹x ≤ b} where a € Rd and be R. Show that halfspaces are convex. Using (a) and (d), show that {x: c≤a¹x ≤ b} is convex when c < b. (f) Using (b) and (e), show that the d-dimensional cube {x: 0≤ i ≤ 1 for i = {1,...,d}} is convex

3. Use the definition of convex sets to answer the following: Show that if the sets S and T are convex, then SOT is convex. Show that the intersection of any number of convex sets is convex. A hyperplane in Rd is a set of points of the form {r: ax=b} where a € Rd and b E R. Show that hyperplanes are convex. Hint: If you're having trouble seeing why this is true in general, try the problem with a simple concrete example in 2-dimensions. La (d) A halfspace Rd is a set of points of the form {x: a¹x ≤ b} where a € Rd and be R. Show that halfspaces are convex. Using (a) and (d), show that {x: c≤a¹x ≤ b} is convex when c < b. (f) Using (b) and (e), show that the d-dimensional cube {x: 0≤ i ≤ 1 for i = {1,...,d}} is convex

Elementary Geometry for College Students
Elementary Geometry for College Students
6th Edition
ISBN:9781285195698
Author:Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:Daniel C. Alexander, Geralyn M. Koeberlein
Chapter10: Analytic Geometry
Section10.CT: Test
Problem 22CT
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ONLY write the （d）(e) (f) 

3. Use the definition of convex sets to answer the following: Show that if the sets S and T are convex, then SnT is convex. Show that the intersection of any number of convex sets is convex. A hyperplane in Rd is a set of points of the form {r: ax=b} where a € Rd and b E R. Show that hyperplanes are convex. Hint: If you're having trouble seeing why this is true in general, try the problem with a simple concrete example in 2-dimensions. La (d) A halfspace Rd is a set of points of the form {x: a¹x ≤ b} where a € Rd and be R. Show that halfspaces are convex. Using (a) and (d), show that {x: c≤a¹x ≤ b} is convex when c < b. (f) Using (b) and (e), show that the d-dimensional cube {x: 0≤x≤ 1 for i = {1,...,d}} is convex
Transcribed Image Text:3. Use the definition of convex sets to answer the following: Show that if the sets S and T are convex, then SnT is convex. Show that the intersection of any number of convex sets is convex. A hyperplane in Rd is a set of points of the form {r: ax=b} where a € Rd and b E R. Show that hyperplanes are convex. Hint: If you're having trouble seeing why this is true in general, try the problem with a simple concrete example in 2-dimensions. La (d) A halfspace Rd is a set of points of the form {x: a¹x ≤ b} where a € Rd and be R. Show that halfspaces are convex. Using (a) and (d), show that {x: c≤a¹x ≤ b} is convex when c < b. (f) Using (b) and (e), show that the d-dimensional cube {x: 0≤x≤ 1 for i = {1,...,d}} is convex
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