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Extra proofs of some Results. Version: 19-10-2011 (Georg Still)
 

Summary: Extra proofs of some Results.
Version: 19-10-2011 (Georg Still)
CO, Chapter0,1 p 1
Th.1.19 [Krein­Milman Theorem]
Let C Rn be a compact convex set. Then C is the convex hull
of its extreme points.
Proof By using
L.1.18 and the following
Th.2.23 Let = U Rn be convex and w / U. Then
there exists a separating hyperplane H = {x | aT x = },
0 = a Rn, R such that
aT
w aT
x x U
and > aT u0 for some u0 U.
CO, Chapter0,1 p 2
Proof. By induction on k := dim C (= dim aff (C))
k=1: Then C = {c}, a singleton, and the result is true.
k - 1 k: Define
K := conv {x | x is extreme point of C} C. Then = K,

  

Source: Al Hanbali, Ahmad - Department of Applied Mathematics, Universiteit Twente

 

Collections: Engineering