Let the mean width of the convex set $L\subset \mathbb{R}^n$ defined by $$ b(L):=\frac{2}{\omega_{n}}\int_{S^{n-1}} {h(L,v)} \, d \mathcal{H}^{n-1}(v),$$ where $\mathcal{H}^{n-1}(\cdot)$ denotes the $(n-1)$-dimensional Hausdorff measure (i.e. the surface area measure), $\omega_n$ is the surface area of the unit sphere $ S^{n-1}=\{x\in R^{n} : \|x\|=1\}$, and $h(L,.)=h_L:\mathbb{R}^n\mapsto \mathbb{R}$ is the support function $$ h(L,v)=\max\left\{\langle x,v\rangle \mid x\in L \right\}, \quad \text{for all } v\in\mathbb{R}^n .$$
How can one show, from the definition above, that if $P$ is a polytope with edges $e_1,\ldots,e_k$, then $$b(P)= c_n \sum_{i=1}^{k} \ell_i \theta_i ,$$ where $\ell_i$ is the length of the edge $e_i$ and $\theta_i$ is the external angle of $P$ at the edge $e_i$, and $c_n$ a constant depending only on the dimension $n$?