For the d-shadow on grids [k]^n, initial segments of a colex-type order minimize shadow size, and the same holds for layers with exactly r non-zero coordinates.
Kleitman, Extremal hypergraph problems
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A grid generalisation of the Kruskal-Katona theorem
For the d-shadow on grids [k]^n, initial segments of a colex-type order minimize shadow size, and the same holds for layers with exactly r non-zero coordinates.