Mathematics basis I

Affine set

A line through point x1 and point x2 can be be expressed as follows

(1)

In Euclidean space, an affine set is such a set that contains the line through any two distinct points in the set, e.g., the solution set of linear equations as follows

(2)

To verify that, supposing we have solutions x1 and x2  as follows

(3)

 such that we have 

(4)

Note that every affine set can be also expressed as solution set of system of linear equations

Convex set

In Euclidean space, convex set is such a set that contains line segment between any two points in the set, which can be expressed as follows

(5)

One can readily find that convex set is a subset of affine set. If the set does not contain all the line segments, it is called concave. Examples of convex, and concave sets are given in fig.1

Fig.1 convex & concave (Ref.)

Convex hull

Assuming that we have a set S=[x1, x2, …, xn]. Then the convex combination of its k point, i.e., x1, x2, …, xk, can be defined as follows.

(6)

Convex hull is actually the set that contains all convex combinations of points in S. In other words, convex hull is the smallest set that includes set S. Therefore, convex hull can be also called convex envelope or convex closure .( See Fig. 2 and Fig. 3)

Fig. 2 convex set and convex hull

Fig. 3 convex hull (Ref.)

Convex cone

Conic combination of x1 and x2 can bedefined as follows

(7)

Convex cone is the set that contains all conic combinations of points in the set. Namely, we call a set C a convex cone if any non-negative combination of elements from C remains in C.

Fig. 4 convex cone