A bounded plane convex region symmetric about a lattice point and with area must contain at least three lattice points in the interior. In dimensions, the theorem can be generalized to a region with area , which must contain at least three lattice points. The theorem can be derived from Blichfeldt's theorem.
Minkowski Convex Body Theorem
See alsoBlichfeldt's Theorem, Pick's Theorem
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ReferencesBorwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 17, 2003.Hilbert, D. and Cohn-Vossen, S. "Minkowski's Theorem." §6.3 in Geometry and the Imagination. New York: Chelsea, pp. 41-44, 1999.Minkowski, H. Geometrie der Zahlen. Leipzig, Germany: Teubner, 1912.Olds, C. D.; Lax, A.; and Davidoff, G. The Geometry of Numbers. Washington, DC: Math. Assoc. Amer., 2000.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 99, 1999.Warmus, W. Colloq. Math. I 1, 45-46, 1947.
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Weisstein, Eric W. "Minkowski Convex Body Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MinkowskiConvexBodyTheorem.html