Doignon's theorem in geometry is an analogue of Helly's theorem for the integer lattice. It states that, if a family of convex sets in d {\displaystyle d} -dimensional Euclidean space have the property that the intersection of every 2 d {\displaystyle 2^{d}} contains an integer point, then the intersection of all of the sets contains an integer point.
Doignon's theorem in geometry is an analogue of Helly's theorem for the integer lattice. It states that, if a family of convex sets in d {\displaystyle d} -dimensional Euclidean space have the property that the intersection of every 2 d {\displaystyle 2^{d}} contains an integer point, then the intersection of all of the sets contains an integer point.
Source: Wikipedia "Doignon's theorem" · CC BY-SA 4.0
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