# Separation and Support Theorems

• Michael J. Panik
Chapter
Part of the Theory and Decision Library book series (TDLB, volume 24)

## Abstract

In section 4 of Chapter 1 we encountered the definitions of a hyperplane and a half-plane (-space). Let us now examine these concepts in greater detail. In particular, we shall define a variety of different types of hyperplanes which will be of considerable importance in what follows. To briefly review these definitions, let us note first that for a vector C(# 0)∈ R n and a scalar a ∈ R, a hyperplane
$$H = \left\{ {x\left| {C'x = \alpha ,x \in {R^n}} \right.} \right\}$$
is an (n−1) — dimensional linear variety or affine set. Next, any hyperplane ℌ in R n generates the two closed half-spaces
$$\begin{gathered} \left\{ {{H^ + }} \right\} + \left\{ {x\left| {C'x \geqslant \alpha ,x \in {R^n}} \right.} \right\}, \hfill \\ \left\{ {{H^ - }} \right\} + \left\{ {x\left| {C'x \leqslant \alpha ,x \in {R^n}} \right.} \right\}, \hfill \\ \end{gathered}$$

## Keywords

Boundary Point Separation Theorem Nonempty Convex Supporting Hyperplane Strong Separation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.