  The feasible region $\Omega_{{\rm ind}}(F)$ of a graph $F$ is the collection
of points $(x,y)$ in the unit square such that there exists a sequence of
graphs whose edge densities approach $x$ and whose induced $F$-densities
approach $y$. A complete description of $\Omega_{{\rm ind}}(F)$ is not known
for any $F$ with at least four vertices that is not a clique or an independent
set. The feasible region provides a lot of combinatorial information about $F$.
For example, the supremum of $y$ over all $(x,y)\in \Omega_{{\rm ind}}(F)$ is
the inducibility of $F$ and $\Omega_{{\rm ind}}(K_r)$ yields the Kruskal-Katona
and clique density theorems.
  We begin a systematic study of $\Omega_{{\rm ind}}(F)$ by proving some
general statements about the shape of $\Omega_{{\rm ind}}(F)$ and giving
results for some specific graphs $F$. Many of our theorems apply to the more
general setting of quantum graphs. For example, we prove a bound for quantum
graphs that generalizes an old result of Bollob\'as for the number of cliques
in a graph with given edge density. We also consider the problems of
determining $\Omega_{{\rm ind}}(F)$ when $F=K_r^-$, $F$ is a star, or $F$ is a
complete bipartite graph. In the case of $K_r^-$ our results sharpen those
predicted by the edge-statistics conjecture of Alon et. al. while also
extending a theorem of Hirst for $K_4^-$ that was proved using computer aided
techniques and flag algebras. The case of the 4-cycle seems particularly
interesting and we conjecture that $\Omega_{{\rm ind}}(C_4)$ is determined by
the solution to the triangle density problem, which has been solved by
Razborov.
