Pointed cone

Remarks

Equivalently, $C$ is nonempty and $\alpha x + \beta y \in C$ for all $x, y \in C$ and all real $\alpha, \beta \ge 0$. In this form a pointed cone is a submodule of $V$ over the semiring of nonnegative reals.

Example: the closed upper half-plane $H = \set{(x, y) \in \R^2 : y \ge 0}$. It is closed under sums and positive multiples, and it contains $(0, 0)$. It contains the whole $x$-axis, so it is pointed in this sense but not salient.

Non-example: the open quadrant $\set{(x, y) : x > 0, y > 0}$ is a convex cone, but it does not contain $(0, 0)$.

Warning on terminology: many texts on convex analysis say pointed for a convex cone $C$ with $C \cap (-C) \subseteq \set{0}$, that is, a cone that contains no line. That property is called salient here.