Hyperplane Separation Theorem
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A bit of context to put this on my stats blog: I’m reading Real Analysis books again as a part of my studies. I used to visit Kim C. Border site from time to time to read his excellent materials, and now I read that he passed away. I never audited one of his courses nor studied at Caltech, but we exchanged several emails from 2012 to 2019, mostly about Linear Algebra, and lately also about Arrow’s Impossibility Theorem and social debates in a moment when Chile entered a political crisis that we still haven’t solved. The proof of this theorem, heavily inspired from his style, is a way to tribute him as a very positive influence during my economics studies.
Theorem (Hyperplane Separation Theorem). Let A and B two convex, disjoint, non-empty subsets of Rn. If A is closed and B is compact, exists p∈Rn∖{0} such that p⋅a<p⋅b∀(a,b)∈A×B.
Proof. Shall be made under a “divide and conquer” approach.
If A is closed, define the function
f:B→Rb↦mina∈A‖b−a‖.
If B is compact, exists b0∈B such that f(b0)≤f(b)∀b∈B.
Let y∈A such that f(b0)=‖b0−y‖. The vector
p=b0−y‖b0−y‖
From p⋅p>0, it follows that
p⋅b0−y‖b0−y‖>0,
From (*) we need to prove that p⋅b0≤p⋅b and p⋅a<p⋅y.
Fix a∈A and define the function
g:[0,1]→Rλ↦‖b0+y–λ(a−y)‖2.
Fix b∈B and define the function
h:[0,1]→Rλ↦‖b0−y–λ(b−y)‖2.
Finally, (*), (**) and (***) lead to conclude that p⋅a<p⋅b.
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