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Let $P$ and $Q$ be two convex lattice polygons in $\mathbb{R}_+^2$ and let $P+Q$ be their Minkowski sum. Given a set $S \subset \mathbb{R}^2$, we let $L(S) =\#( S \bigcap \mathbb{Z}^2)$.

The equality $L(\partial (P+Q)) = L(\partial P) + L(\partial Q)$ seems to hold in many cases (for the Minkowski sum of rectangles for example) but not in any case (for instance, it is not true for the Minkowski sum of the unit square and the segment with vertices $(0,0)$ and $(2,0)$).

I am trying to find sufficient and necessary conditions on $P$ and $Q$ for the equality to be valid, but I don't even know what kind of conditions I should look for. Do you have any suggestions ?

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  • $\begingroup$ Segments seem to cause trouble. For non-segments, it seems like there is a bijection between the facets of P+Q, and the union of the facets of P and of Q. (Though if P and Q both have facets with the same outer normal, they form one segment in P+Q.) $\endgroup$
    – Hugh Thomas
    Commented Nov 6 at 22:30

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