Cone

A set $$C$$ is called a cone if for every $$x \in C$$ and $$\theta \ge 0$$ we have $$\theta x \in C$$. Cones are not necessarily convex. Cones can have very unintuitive shapes. For example, a line that passes from the origin is a cone. Also you can rotate the line, and the area it swipes will also form a cone. See some cone shapes in Fig. 35-41.

Proper Cone
Proper cones are a subset of cones that have more intuitive shapes. Formally, a proper cone $$K \subseteq \mathbf{R}^n$$ satisfies the following propeties
 * Is convex
 * Is closed
 * Is solid (i.e., it has nonempty interior)
 * is pointed, which means that it contains no line (or, equivalently, $$x \in K$$, $$-x \in K \implies x = 0$$)

What's the point of cones?
The concept of cones is central for defining generalized inequalities. For example, the componentwise inequality for two vectors $$x$$ and $$y$$, namely $$x \preceq y$$, implies that the vector $$y-x$$ belongs to the (proper) cone