How to prove a set of vectors $\{x\in \mathbb{R}^n: \| x\|<1\}$ is polytope?

Question

Assume we have a set of vectors $\{x\in \mathbb{R}^n: \| x\|<1\}$, how can we should that this is a polytope?

Could we use the property that a convex polytope may be defined as an intersection of a finite number of half-spaces?

Answer 1

For $y\in\mathbb R^n$ with $\Vert y\Vert=1$, let $H_y=\{x\in\mathbb R^n\mid \langle x,y\rangle<1\}$, which is a half-space. Then $B:=\{x\in\mathbb R^n\mid\Vert x\Vert<1\}$ can be written as $$ B=\bigcap_{\Vert y\Vert=1} H_y.$$

Proof: If $x\in B$ and $\Vert y\Vert=1$ then $\langle x,y\rangle\le \Vert x\Vert\cdot\Vert y\Vert=\Vert x\Vert <1$, hence $x\in H_y$. If $x\notin B$, let $y=\frac1{\Vert x\Vert}x$. Then $\Vert y\Vert =1$ and $\langle x,y\rangle=\frac1{\Vert x\Vert}\langle x,x\rangle = \Vert x\Vert\ge 1$, hence $x\notin H_y$. $_\square$