Given an $n$-dimensional polytope $P\subset \mathbb{R}^n$.

For a subset of indices $i_1,\ldots,i_k$ of $\{1,\ldots,n\}$ and reals $a_1,\ldots,a_k$, we denote by $P(x_{i_1}=a_{1},\ldots,x_{i_k}=a_{k})$ the set of point of the polytope $P$ where coordinates $x_{i_j}$ are fixed to some constant $a_j$ (it is the intersection of $P$ with the hyperplanes of equations $x_{i_j}=a_{j}$). Let $f(a_1,\ldots,a_k)$ the $n-k$-volume of $P(x_{i_1}=a_{1},\ldots,x_{i_k}=a_{k})$. I think that this function is piecewise polynomial in the variables $a_j$ and that this function is continuous in its support (i.e. the set where it is not null). These two results seems quite elementary but I didn't find a reference nor a proof.