  In this paper extremal problems for uniform hypergraphs are studied in the
general setting of hereditary properties. It turns out that extremal problems
about edges are particular cases of a general analyic problem about a recently
introduced graph parameter. The paper builds a basis for the systematic study
of this parameter and illustrates a range of various proof tools. It is shown
that extremal problems about the number of edges of uniform hypergraphs are
asymptotically equivalent to extremal problems about the largest eigenvalue;
this result is new even for 2-graphs. Several concrete problems are adressed
and solutions to many more are suggested. A number of open problems are raised
and directions for further studies are outlined.
