Definition: Cone and cone order

Let $V$ be a finite-dimensional real vector space. A cone $K\subset V$ is proper if it is closed, convex, pointed, $K\cap (-K)=\{0\}$, and has nonempty interior in $V$. Its translation-invariant order is

For $m\ge 3$, the closed Lorentz cone is

The statement that $K$ is linearly isomorphic to a Lorentz cone means that an invertible linear map $T:V\to {\mathbb{R}}^m$ satisfies $T(K)={\mathcal{L}}_m$.