11. Open and Closed Half Spaces
A hyperplane divides the whole space E n into three mutually disjoint sets given by
X1 = { x: cx > z }
X2 = { x: cx = z }
X3 = { x: cx < z }
The sets x1 and x2 are called‘ open half spaces’. The sets { x: cx ≤ z } and { x: cx ≥ z } are called‘ closed half spaces’.
12. Parallel Hyperplanes Two hyperplanes c1x = z1 and c2x = z2 are said to be parallel, if they have the same unit normals i. e., if c1 = Xc2 for λ, λ being non-zero.
13. Convex Combination A convex combination of a finite number of points x1, x2,…., xn is defined as a point x = λ1 x1 + λ2x2 + …. + λnxn, where λi is real and ≥ 0, ∀ and
14. Convex Set A set of points is said to be convex, if for any two points in the set, the line segment joining these two points is also in the set.
or
A set is convex, if the convex combination of any two points in the set, is also in the set.
15 Extreme Point of a Convex Set A point x in a convex set c is called an‘ extreme point’, if x cannot be expressed as a convex combination of any two distinct points x1 and x2 in c.
16. Convex Hull The convex hull c( X) of any given set of points X is the set of all convex combinations of sets of points from X.