Minkowski: volume forces a nonzero integer lattice point

What changed

Minkowski used translates of a half-scale centrally symmetric convex body to prove M^3 I<=8 for the minimum nonzero integer-lattice gauge M and unit-body volume I in three dimensions. Consequently a centered closed convex body of volume at least8 contains a nonzero integer lattice point. He connected the geometric bound with linear forms and simultaneous rational approximation.

Scientific scope

Minkowski used translates of a half-scale centrally symmetric convex body to prove M^3 I<=8 for the minimum nonzero integer-lattice gauge M and unit-body volume I in three dimensions. Consequently a centered closed convex body of volume at least8 contains a nonzero integer lattice point. He connected the geometric bound with linear forms and simultaneous rational approximation. The 1896 authorized French translation by L. Laugel presents the argument for Z^3 with a positive finite convex gauge and central symmetry; arbitrary nonsymmetric sets or an open body at equality are not covered. The author states that his book removes the three-variable restriction, but that general-dimensional proof lies outside this cited treatment. Later successive-minima or unrestricted lattice versions are not silently imported.

Evidence

The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The 1896 authorized French translation by L. Laugel presents the argument for Z^3 with a positive finite convex gauge and central symmetry; arbitrary nonsymmetric sets or an open body at equality are not covered. The author states that his book removes the three-variable restriction, but that general-dimensional proof lies outside this cited treatment. Later successive-minima or unrestricted lattice versions are not silently imported.

Contributors

• The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.

• Sur les proprietes des nombres entiers qui sont derivees de l'intuition de l'espace (authorized French translation)

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