Convex cone

Remarks

Equivalently, $C$ is closed under addition and under multiplication by positive scalars. Equivalently again, $C$ is convex and $t x \in C$ for all $x \in C$ and $t > 0$.

Example: the nonnegative orthant $\R^n_{\ge 0} = \set{x \in \R^n : x_i \ge 0 \text{ for all } i}$. If $x, y$ have nonnegative coordinates and $\alpha, \beta > 0$, then each coordinate $\alpha x_i + \beta y_i$ is nonnegative.

Non-example: the closed unit disc in $\R^2$ is convex, but $(1, 0)$ lies in it and $2 \cdot (1, 0) = (2, 0)$ does not.

Conventions differ: many authors also allow $\alpha = \beta = 0$, so that every nonempty convex cone contains $0$. A convex cone that contains $0$ is called a pointed cone in some texts. The same definition works over any ordered field in place of $\R$.