Supporting hyperplane

The separation theorem (also set by Eidelheit ) is a mathematical theorem about the possibilities for the separation of convex sets in normed vector spaces (or more generally locally convex spaces ) by linear functionals. It is geometric consequences of the Hahn- Banach.

First formulation

The simplest version of the separation theorem is as follows:

Be a normed vector space (or locally convex space ) or over. Be further a closed convex set and. Then there exists a continuous linear functional with

Here, the real part and the dual space of. We then say: The functional separates the point from the crowd.

Additional formulations

In the above formulation, the point may be replaced by a compact convex set. One then obtains the following sentence:

Be a normed vector space (or locally convex space ) or over. Be forward a non - empty, closed, convex set and a non-empty, compact, convex set. Then there exists a continuous linear functional with

Finally, we come to a weaker separation property if you omitted in the above version to the seclusion and compactness:

Be a normed vector space (or locally convex space ) or over. Be more non-empty, disjoint, convex sets, be open. Then there exists a continuous linear functional with

Hyperplanes

Sets of the form, and wherein are completed hyperplanes. Disassemble the X space into an upper half-space and a lower half-space. For a compact convex set and a disjoint closed convex set to one by the above separation theorem can find a hyperplane, so that the two quantities are in different half-spaces, respectively inside this half-spaces. We say that the hyperplane separating the two convex sets. This is especially clear in the 2-dimensional and 3 -dimensional case, because in these cases the hyperplanes or straight planes.

If one has two disjoint convex sets in, one of which is open, so there is this after the latter version of the separation theorem also a hyperplane so that the two quantities are in different half-spaces. In general, but you can not reach that both lie in the interior of the half-spaces. To Consider in the lower half plane, and the open volume above the graph of the exponential function. As illustrated by drawing on the left, is the only separating hyperplane, and not in the interior of the corresponding half-space.

Applications

This theorem has many important applications outside of functional analysis and provides much evidence for a non- constructive existence argument is, among other things:

• Existence of Subdifferentialen for suitable formulated generalized directional derivatives.
• Proof of Farkas ' Lemma, ie application in convex optimization.
• Proof of the fundamental theorem of the theory of pricing fair pricing of derivatives in the multi-period model.