  Discrete strip-concave functions considered in this paper are, in fact,
equivalent to an extension of Gelfand-Tsetlin patterns to the case when the
pattern has a not necessarily triangular but convex configuration. They arise
by releasing one of the three types of rhombus inequalities for discrete
concave functions (or ``hives'') on a ``convex part'' of a triangular grid. The
paper is devoted to a combinatorial study of certain polyhedra related to such
functions or patterns, and results on faces, integer points and volumes of
these polyhedra are presented. Also some relationships and applications are
discussed.
  In particular, we characterize, in terms of valid inequalities, the
polyhedral cone formed by the boundary values of discrete strip-concave
functions on a grid having trapezoidal configuration. As a consequence of this
result, necessary and sufficient conditions on a pair of vectors to be the
shape and content of a semi-standard skew Young tableau are obtained.
