Projections and angle sums of permutohedra.

Let $(x_1,\dots,x_n)\in\mathbb R^n$. Permutohedra of types $A$ and $B$ are convex polytopes in $\mathbb R^n$ defined by $$ \mathcal{P}_n^A=\text{conv}\{(x_{\sigma(1)},\dots,x_{\sigma(n)}):\sigma\in\text{Sym}(n)\} $$ and $$ \mathcal{P}_n^B=\text{conv}\{(\varepsilon_1x_{\sigma(1)},\dots,\varepsilon_nx_{\sigma(n)}):(\varepsilon_1,\dots,\varepsilon_n)\in\{\pm\}^n,\sigma\in\text{Sym}(n)\}, $$ where $\text{Sym}(n)$ denotes the group of permutations of the set $\{1,\dots,n\}$. We derive a closed formula for the number of $j$-faces of $G\mathcal{P}_n^A$ and $G\mathcal{P}_n^B$ for a linear map $G:\mathbb R^n\to\mathbb R^d$ satisfying some minor general position assumptions. In particular, we will show that the face numbers of the projected permutohedra do not depend on the linear mapping $G$. Furthermore, we derive formulas for the sum of the conic intrinsic volumes of the tangent cones of $\mathcal{P}_n^A$ and $\mathcal{P}_n^B$ at all of their $j$-faces. The same is done for the Grassmann angles. We generalize all these results to the class of belt polytopes, which are polytopes whose normal fan is the fan of some hyperplane arrangement.