  The Tur\'an hypergraph problem asks to find the maximum number of $r$-edges
in a $r$-uniform hypergraph on $n$ vertices that does not contain a clique of
size $a$. When $r=2$, i.e., for graphs, the answer is well-known and can be
found in Tur\'an's theorem. However, when $r\geq 3$, the problem remains open.
We model the problem as an integer program and call the underlying polytope the
Tur\'an polytope. We draw parallels between the latter and the stable set
polytope: we show that generalized and transformed versions of the web and
wheel inequalities are also facet-defining for the Tur\'an polytope. We also
show clique inequalities and what we call doubling inequalities are
facet-defining when $r=2$. These facets lead to a simple new polyhedral proof
of Tur\'an's theorem.
