Convex and Concave Functions

Let us first start with a formal definition of a 2 vector convex combination. Then we will break down the definition into parts and analyze the definition. Then we will formally define and analyze a convex combination with a finite number of vectors in the same manner.

A subset \(S \subseteq \mathbb{R}^{n}\) is said to be a convex set if:

\(\lambda x + (1 - \lambda)y \in S \;\forall \lambda \in [0,1], \forall x,y \in S\)

Our formal definition here is for a convex combination with 2 vectors. Vectors \(x\) and \(y\) are elemenets of the subset \(S\). \(S\) is a proper subset of the real numbers to \(n\) dimensions. Our variable lambda \(\lambda\) has the value betweewn 0 and 1. The result of \(\lambda x + (1 - \lambda)y\) should be an element in \(S\).

Now let us formally define a convex combination with any finite number of vectors. Let us say that \(S \subseteq \mathbb{R}^{n}\) and \(x_{1},...,x_{k}\in S\). Then a vector of the form:

\(y=\sum_{i=1}^{k}\lambda_{i}x_{i} \geq 0 \; \forall i, \sum_{i=1}^{k} \lambda_{i}=1\)

is called a convex combination of the vectors \(x_{1},...,x_{k}\). Here the summation of every \(\lambda_{i}\) is equal to 1 and every \(\lambda_{i}\) is greater is greater than or equal to 0.