## Geometry of numbers |

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absolute value belongs to G boundary point bounded set column completes the proof components Consider contains contradicts convex body defined convex set coordinates corresponding defined by f(x definite quadratic form denote determinant elements equal equation equivalent Euclidean space finite number follows form a basis form a lattice fundamental region gauge function given greatest common divisor greatest lower bound implies independent vectors inner point integers integral vector intersection lattice of g-points lattice point Lecture Lemma limit point linear forms linear subspace minimal vectors Minkowski's First Theorem Minkowski's Theorem n-dimensional non-singular matrix non-trivial solution Note obtained octahedron overlap parquet periods polynomial positive numbers positive-definite possible primitive vector Prom prove Theorem quadratic form reduced reduction conditions satisfies shew space star domain Suppose Theorem XXIX trivial uni-modular matrix uni-modular transformation variables vector g vector group vector in G volume whese