Proper cone

Remarks

A nonempty closed convex cone contains $0 = \lim_{t \to 0^+} t x$, so a proper cone is a pointed cone.

Example: the nonnegative orthant $\R^n_{\ge 0}$. It is a closed convex cone, its interior is the set of vectors with all coordinates positive, and if $x$ and $-x$ both have nonnegative coordinates then $x = 0$.

Non-example: the closed half-plane $\set{(x, y) \in \R^2 : y \ge 0}$ satisfies clauses 1 and 2, but it contains $(1, 0)$ and $(-1, 0)$, so clause 3 fails.

A proper cone $K$ gives a partial order on $\R^n$ by $x \preceq_K y$ if $y - x \in K$. Conventions vary: some sources, Mathlib among them, call any closed pointed cone proper and drop clauses 2 and 3.