Convex Sets and Cones
A set is convex if it does not contain any two points such that the line segment connecting them contains a point not in the set. For example a circle is a convex set. A star is not a convex set, since the points between two tips of the star lie outside of the star. Convex sets can be bounded or unbounded. A circle is a bounded convex set. $R^n$ is an unbounded convex set.
A cone is type of convex set defined as
\[\{Ax|x \geq 0\}\]It is literally shaped like a cone. Take the cone created by vectors $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. The cone spanned by these two vectors looks like this:
Non-negative Orthant
The non-negative orthant is the simplest cone. It is defined as
\[\{x | x \geq 0\}\]In $R^2$ this is simply the first quadrant.