= Solution
Let $c(A)=[n]\setminus A$ and reverse the order of the ground set by $j\mapsto n+1-j$. The resulting bijection sends <lexicographic order> on $[n]^{(r)}$ to <colexicographic order> on $[n]^{(n-r)}$. It also sends the iterated <upper shadow> of an $r$-uniform family at level $r+1$ to the <lower shadow> of the corresponding $(n-r)$-uniform family, up to the same harmless reversal of the ground set.
The <Kruskal-Katona theorem> says that this lower shadow is smallest for an initial colex segment. Undoing the complement and reversal therefore says that the upper shadow of a family of fixed size in $[n]^{(r)}$ is smallest for the initial lex segment.
Solved by gpt-5.6-sol high.
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