Norm cone is a proper cone

Question

For a finite vector space $H$ define the norm cone $K = \{ (x, \lambda) \in H \oplus \mathbb{R} : \lVert x \rVert \le \lambda \}$ where $\lVert x \rVert$ is some norm. There are endless lecture notes pointing out that this is a convex cone (as the pre-image of a convex set under the perspective function). In fact, I believe it is a proper cone. Since the proof of that is somewhat tedious, is there some reference I can cite for this fact?

Answer 1

Obviously $K$ is convex and closed as it is defined by the inequality $g(x,\lambda):=\|x\| - \lambda\le0$ with $g$ continuous and convex. Also it has non-empty interior. Take $x_0\ne0$, $\lambda_0>\|x_0\|$. Then $\|x\| < \lambda$ in a neighborhood of $(x_0,\lambda_0)$. If $(x,\lambda),(-x,-\lambda) \in K$, then $0\le \|x\| \le \min(\lambda,-\lambda)$ implying $\lambda=0$ and so $x=0$. And $K$ is pointed.