Email Gateway (Aug 22 2022 at 15:46): Email Gateway (Aug 22 2022 at 15:46):From: Gerwin Klein <gerwin.klein@data61.csiro.au>
https://www.isa-afp.org/entries/Minkowskis_Theorem.shtml
Minkowski's Theorem
by Manuel Eberl
Minkowski's theorem relates a subset of ℝ^n, the Lebesgue measure, and the integer lattice ℤ^n: It states that any convex subset of ℝ^n with volume greater than 2^n contains at least one lattice point from ℤ^n\{0}, i. e. a non-zero point with integer coefficients.
A related theorem which directly implies this is Blichfeldt's theorem, which states that any subset of ℝ^n with a volume greater than 1 contains two different points whose difference vector has integer components.
The entry contains a proof of both theorems.
Enjoy!
Gerwin
Email Gateway (Aug 22 2022 at 15:47):From: Gerwin Klein <kleing@unsw.edu.au>
https://www.isa-afp.org/entries/Minkowskis_Theorem.shtml
Minkowski's Theorem
by Manuel Eberl
Minkowski's theorem relates a subset of ℝ^n, the Lebesgue measure, and the integer lattice ℤ^n: It states that any convex subset of ℝ^n with volume greater than 2^n contains at least one lattice point from ℤ^n\{0}, i. e. a non-zero point with integer coefficients.
A related theorem which directly implies this is Blichfeldt's theorem, which states that any subset of ℝ^n with a volume greater than 1 contains two different points whose difference vector has integer components.
The entry contains a proof of both theorems.
Enjoy!
Gerwin
Last updated: Nov 21 2024 at 12:39 UTC