Volume of intersection of a convex polytope with an affine space. 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.
 A: Thank you for your comments.
Putting all the things together we have an answer.
The proof for the piecewise polynomial part is given by Pietro (but not the continuity).
The concavity of $f^{1/n-k}$ gives the continuity (at least in the interior of the polytope which is sufficient for me, today). This concavity is due to the so called Brunn-Minkowski inequality  (and the function itself is unimodal) see the proof (for only one hyperplane) in a good reference for my problem : "On the sectional area of convex polytope" by David Avis et al. For your information, in dimension 3 they show that the function is piecewise quadratic using drums...
If you have a more direct answer, please don't hesitate to share it. 
A: Following Andreas Blass's hints, one possibility to show the continuity is the following. Let $d:=n-k$ be the dimension of the cutting affine spaces. 
First, let's say that a face of a polytope is a bad face  if it spans an affine space containing a $d$ dimensional subspace parallel to the cutting space; otherwise, call it a nice face. If all faces of a polytope are nice, let's say it's a nice polytope, after all.
For a nice simplex, one checks that the function  $f$  is continuous on the whole $a$-space. 
By genericity, any polytope can be subdivided into nice simplexes, plus a number of simplexes whose bad faces  are included into some bad faces of the polytope.
In other words,  a generic triangulation produces a simplicial complex where no bad faces are introduced. This shows that the points of discontinuity of the function $f$ of the polytope are exactly the projection in the $a$-space of its bad faces.
