# Affine Vs Convex

Affine Vs Convex. We will write this functionally as: Let cbe a convex set and x= p n i=1 ix

A convex combination of elements of ais any nite sum of the form (1) with i 0 (i= 1;:::;n). A set c is convex if the line segment between any two points in c lies in c. Empty and singleton sets are both affine and convex set.

### Let Cbe A Convex Set And X= P N I=1 Ix

+ c n x n, so an affine function would be defined as: There is equality in the right hand inequality if and only if μ is discrete or πl is a euclidean ball. Hence, every affine set is also a convex set.

### S Is A Cone If It Is Closed Under Nonnegative Scalar Multiplication.

So doesn't that make the definition of convex sets more restricted than affine sets? In addition, an affine function is sometimes defined as a linear form plus a number. Every convex/a ne/linear set in a vector space xis closed under making convex/a ne/linear combinations of its elements.

### The Convex Hull Of Two Points, For Example, Comprises Those Points And The Line Segment Between Them.

Different from the definition of affine sets, in a convex set we only need the segment lies on it, which characterized by the range of λ. Let s be a subset of. A set $s$ is convex iff for every pair of points $x,y\in s$, the line segment $\overline{xy}$ joining $x$ to $y$ is a subset of $s$.

### A Convex Set Lies Inside An Affine Set;

A set is convex if whenever it contains two points, it contains every point of the line segment joining those two points. Any subspace is a convex set. C 1 x 1 +.

### (6) Therefore, The Cost Function Would Be Equivalent To #Α C # −1 1 \$ # 2 R = Α C!

Notice that are non negative. A linear form has the format c 1 x 1 +. Both of these half spaces are convex.