  In this paper we settle long-standing questions regarding the combinatorial
complexity of Minkowski sums of polytopes: We give a tight upper bound for the
number of faces of a Minkowski sum, including a characterization of the case of
equality. We similarly give a (tight) upper bound theorem for mixed faces of
Minkowski sums. This has a wide range of applications and generalizes the
classical the Upper Bound Theorems of McMullen and Stanley.
  Our main tool is relative Stanley--Reisner theory, a powerful generalization
of the algebraic theory of simplicial complexes inaugurated by Hochster,
Reisner, and Stanley. A key feature of our theory is the ability to accomodate
topological as well as combinatorial restrictions. We illustrate this by
providing several simplicial isoperimetric and reverse isoperimetric
inequalities.
