  Given a family $S$ of $k$--subsets of $[n]$, its lower shadow $\Delta(S)$ is
the family of $(k-1)$--subsets which are contained in at least one set in $S$.
The celebrated Kruskal--Katona theorem gives the minimum cardinality of
$\Delta(S)$ in terms of the cardinality of $S$. F\"uredi and Griggs (and
M\"ors) showed that the extremal families for this shadow minimization problem
in the Boolean lattice are unique for some cardinalities and asked for a
general characterization of these extremal families.
  In this paper we prove a new combinatorial inequality from which yet another
simple proof of the Kruskal--Katona theorem can be derived. The inequality can
be used to obtain a characterization of the extremal families for this
minimization problem, giving an answer to the question of F\"uredi and Griggs.
Some known and new additional properties of extremal families can also be
easily derived from the inequality.
