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2.1 Let C C R" be a convex set, with x1, ..., xk E C, and let 01,..., Ok E R satisfy 0i > 0, 01 + ...+ Ok = 1. Show that 01x1 + + Okxk E C. (The definition of convexity is that this holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k.

2.1 Let C C R" be a convex set, with x1, ..., xk E C, and let 01,..., Ok E R satisfy 0i > 0, 01 + ...+ Ok = 1. Show that 01x1 + + Okxk E C. (The definition of convexity is that this holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k.

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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Exercises Definition of convexity 2.1 Let C C R" be a convex set, with x1,... , xk E C, and let 01, ... , Ok E R satisfy 0; 2 0, 01 + ...+ Ok = 1. Show that 01x1 + ...+ Okxk E C. (The definition of convexity is that this holds fork = 2; you must show it for arbitrary k.) Hint. Use induction on k. 2.2 Show that a set is convex if and only if its intersection with any line is convex. Show that a set is affine if and only if its intersection with any line is affine. 2.3 Midpoint convexity. A set C is midpoint conver if whenever two points a, b are in C, the average or midpoint (a + b)/2 is in C. Obviously a convex set is midpoint convex. It can be proved that under mild conditions midpoint convexity implies convexity. As a simple case, prove that if C is closed and midpoint convex, then C is convex. 2.4 Show that the convex hull of a set S is the intersection of all convex sets that contain S. (The same method can be used to show that the conic, or affine, or linear hull of a set S is the intersection of all conic sets, or affine sets, or subspaces that contain S.) Examples 2.5 What is the distance between two parallel hyperplanes {x E R" | a"x = b1} and {x E R" | a"x = b2}? 2.6 When does one halfspace contain another? Give conditions under which (where a + 0, ã 7 0). Also find the conditions under which the two halfspaces are equal. 2.7 Voronoi description of halfspace. Let a and b be distinct points in R". Show that the set of all points that are closer (in Euclidean norm) to a than b, i.e., {x | ||x– a||2 < ||x – ||2}, is a halfspace. Describe it explicitly as an inequality of the form c' x < d. Draw a picture. 2.8 Which of the following sets S are polyhedra? If possible, express S in the form S {x | Ax 3 b, Fx = g}. (a) S= {y1a1 + y2a2 | –1< yı < 1, –1 < y2 < 1}, where a1, a2 E R". {x € R" | x E 0, 1"x a1, ... , an E R and b1, b2 e R. (c) S = {x € R" | x E 0, x"y < 1 for all y with ||y||2 = 1}. (d) S = {x € R" | x > 0, x"y<1 for all y with , ly:| = 1}. (b) S = 1, Σα, α, b2}, where :=1
Transcribed Image Text:Exercises Definition of convexity 2.1 Let C C R" be a convex set, with x1,... , xk E C, and let 01, ... , Ok E R satisfy 0; 2 0, 01 + ...+ Ok = 1. Show that 01x1 + ...+ Okxk E C. (The definition of convexity is that this holds fork = 2; you must show it for arbitrary k.) Hint. Use induction on k. 2.2 Show that a set is convex if and only if its intersection with any line is convex. Show that a set is affine if and only if its intersection with any line is affine. 2.3 Midpoint convexity. A set C is midpoint conver if whenever two points a, b are in C, the average or midpoint (a + b)/2 is in C. Obviously a convex set is midpoint convex. It can be proved that under mild conditions midpoint convexity implies convexity. As a simple case, prove that if C is closed and midpoint convex, then C is convex. 2.4 Show that the convex hull of a set S is the intersection of all convex sets that contain S. (The same method can be used to show that the conic, or affine, or linear hull of a set S is the intersection of all conic sets, or affine sets, or subspaces that contain S.) Examples 2.5 What is the distance between two parallel hyperplanes {x E R" | a"x = b1} and {x E R" | a"x = b2}? 2.6 When does one halfspace contain another? Give conditions under which (where a + 0, ã 7 0). Also find the conditions under which the two halfspaces are equal. 2.7 Voronoi description of halfspace. Let a and b be distinct points in R". Show that the set of all points that are closer (in Euclidean norm) to a than b, i.e., {x | ||x– a||2 < ||x – ||2}, is a halfspace. Describe it explicitly as an inequality of the form c' x < d. Draw a picture. 2.8 Which of the following sets S are polyhedra? If possible, express S in the form S {x | Ax 3 b, Fx = g}. (a) S= {y1a1 + y2a2 | –1< yı < 1, –1 < y2 < 1}, where a1, a2 E R". {x € R" | x E 0, 1"x a1, ... , an E R and b1, b2 e R. (c) S = {x € R" | x E 0, x"y < 1 for all y with ||y||2 = 1}. (d) S = {x € R" | x > 0, x"y<1 for all y with , ly:| = 1}. (b) S = 1, Σα, α, b2}, where :=1
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