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 .