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Let B[0, 1] = {x € R" : ||x|| < 1} be the l ball of centre zero and radius 1. Recall from chapter 1 (Beck) that ||x|| = max; r:|. Denote by Extr(B[0, 1]) the set of extreme points of B[0, 1]. Define the set T = {x € R" : |x;| = 1Vj = 1,., n}. Prove the following ... statements. (a) Extr(B.[0,1]) CT. Hint: Show that if x ¢ T then x ¢ Extr(B[0, 1]). Write x = that if x ¢ T and x € B[0, 1], then there must exist a coordinate j such that |r;| < 1. Prove that there exists e > 0 such that both (x1,..., x; + €, ..., xn)' and (x1,..., I; – €, ..., En)' belong to B[0, 1]. Then show that x is a convex combination of the latter (*1, 12,..., Tn)*. Show two vectors. (b) T C Extr(Boo [0, 1]).

Let B[0, 1] = {x € R" : ||x|| < 1} be the l ball of centre zero and radius 1. Recall from chapter 1 (Beck) that ||x|| = max; r:|. Denote by Extr(B[0, 1]) the set of extreme points of B[0, 1]. Define the set T = {x € R" : |x;| = 1Vj = 1,., n}. Prove the following ... statements. (a) Extr(B.[0,1]) CT. Hint: Show that if x ¢ T then x ¢ Extr(B[0, 1]). Write x = that if x ¢ T and x € B[0, 1], then there must exist a coordinate j such that |r;| < 1. Prove that there exists e > 0 such that both (x1,..., x; + €, ..., xn)' and (x1,..., I; – €, ..., En)' belong to B[0, 1]. Then show that x is a convex combination of the latter (*1, 12,..., Tn)*. Show two vectors. (b) T C Extr(Boo [0, 1]).

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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Let B[0, 1] = {x € R" : ||x|| < 1} be the l ball of centre zero and radius 1. Recall from chapter 1 (Beck) that ||x|| = max; r:|. Denote by Extr(B[0, 1]) the set of extreme points of B[0, 1]. Define the set T = {x € R" : |x;| = 1Vj = 1,., n}. Prove the following ... statements. (a) Extr(B.[0,1]) CT. Hint: Show that if x ¢ T then x ¢ Extr(B[0, 1]). Write x = that if x ¢ T and x € B[0, 1], then there must exist a coordinate j such that |r;| < 1. Prove that there exists e > 0 such that both (x1,..., x; + €, ..., xn)' and (x1,..., I; – €, ..., En)' belong to B[0, 1]. Then show that x is a convex combination of the latter (*1, 12,..., Tn)*. Show two vectors. (b) T C Extr(Boo [0, 1]).
Transcribed Image Text:Let B[0, 1] = {x € R" : ||x|| < 1} be the l ball of centre zero and radius 1. Recall from chapter 1 (Beck) that ||x|| = max; r:|. Denote by Extr(B[0, 1]) the set of extreme points of B[0, 1]. Define the set T = {x € R" : |x;| = 1Vj = 1,., n}. Prove the following ... statements. (a) Extr(B.[0,1]) CT. Hint: Show that if x ¢ T then x ¢ Extr(B[0, 1]). Write x = that if x ¢ T and x € B[0, 1], then there must exist a coordinate j such that |r;| < 1. Prove that there exists e > 0 such that both (x1,..., x; + €, ..., xn)' and (x1,..., I; – €, ..., En)' belong to B[0, 1]. Then show that x is a convex combination of the latter (*1, 12,..., Tn)*. Show two vectors. (b) T C Extr(Boo [0, 1]).
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