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
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]).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4fb404b9-8e1c-41d2-a59e-964abbbd4b28%2Fb9044cae-6994-4add-9645-100854e6c622%2Fofwft8x_processed.png&w=3840&q=75)
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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