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In this short note, we establish an edge-isoperimetric inequality for arbitrary product graphs. Our inequality is sharp for subsets of many different sizes in every product graph. In particular, it implies that the 2ᵈ-element sets with smallest edge-boundary in the hypercube are subcubes and is only marginally weaker than the Bollob\'asx2013Leader edge-isoperimetric inequalities for grids and tori. Additionally, it improves two edge-isoperimetric inequalities for products of regular graphs proved by Erde, Kang, Krivelevich, and the first author and answers two questions about edge-isoperimetry in powers of regular graphs raised in their work.
Diskin et al. (Tue,) studied this question.