Doignon's theorem

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 -dimensional Euclidean space have the property that the intersection of every contains an integer point, then the intersection of all of the sets contains an integer point. Therefore, -dimensional integer linear programs form an LP-type problem of combinatorial dimension , and can be solved by certain generalizations of linear programming algorithms in an amount of time that is linear in the number of constraints of the problem and fixed-parameter tractable in its dimension. The same theorem applies more generally to any lattice, not just the integer lattice.

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