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.

翻译：设$S$是$[n]$上的$k$子集族，其下阴影$\Delta(S)$表示至少包含于$S$中一个集合的$(k-1)$子集族。著名的Kruskal–Katona定理给出了$\Delta(S)$的最小基数与$S$基数的关系。Füredi与Griggs（以及Mörs）指出，在布尔格中，该阴影极小化问题的极值族在部分基数下具有唯一性，并请求给出这些极值族的一般刻画。本文证明了一个新的组合不等式，由此可导出Kruskal–Katona定理的另一种简单证明。该不等式可用于刻画该极小化问题的极值族，从而回答了Füredi与Griggs的问题。此外，由该不等式还可轻松推得极值族的若干已知与新的附加性质。