Hyperplane Separation Theorem
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. This theorem, interesting by itself, provides a powerful result used in the different versions and proofs of the finite dimensional supporting hyperplane theorem, the Farkas’ lemma, and the Karush-Kuhn-Tucker theorem. These result are also the basis for the support vector machines algorithm, duality in linear optimization, and many portfolio-related results in finance.
Theorem (Hyperplane Separation Theorem). Let
A consequence of this result, is that any closed and convex set
Proof. Shall be made under a “divide and conquer” approach.
If
If
Let
From
From
Fix
Fix
Finally,
Corollary. Let
, in which case exists such that . , in which case exists such that .
Proof. Again, dividing by parts.
Note. The latter proof assumes that the interior of the adherence of a set is contained within the set. The property is intuitively clear, but only applies for convex sets. As a counter example: Let