  We study the structure of the set of all possible affine hyperplane sections
of a convex polytope. We present two different cell decompositions of this set,
induced by hyperplane arrangements. Using our decomposition, we bound the
number of possible combinatorial types of sections and craft algorithms that
compute optimal sections of the polytope according to various combinatorial and
metric criteria, including sections that maximize the number of $k$-dimensional
faces, maximize the volume, and maximize the integral of a polynomial. Our
optimization algorithms run in polynomial time in fixed dimension, but the same
problems show hardness otherwise. Our tools can be extended to intersection
with halfspaces and projections onto hyperplanes. Finally, we present several
experiments illustrating our theorems and algorithms on famous polytopes.
