Minkowski's second theorem

Let {{mvar|K}} be a closed convex centrally symmetric body of positive finite volume in {{mvar|n}}-dimensional Euclidean space {{math|Rⁿ}}. The gaugeSiegel (1989) p.6 or distanceCassels (1957) p.154Cassels (1971) p.103 Minkowski functional {{math|g}} attached to {{math|K}} is defined by

$$g(x)\; =\; \backslash inf\; \backslash left\backslash \{\backslash lambda\; \backslash in\; \backslash mathbb\{R\}\; :\; x\; \backslash in\; \backslash lambda\; K\; \backslash right\backslash \}\; .$$

Conversely, given a norm {{math|g}} on {{math|Rⁿ}} we define {{mvar|K}} to be

$$K\; =\; \backslash left\backslash \{\; x\; \backslash in\; \backslash R^n\; :\; g(x)\; \backslash le\; 1\; \backslash right\backslash \}\; .$$

Let {{math|Γ}} be a lattice in {{math|Rⁿ}}. The successive minima of {{mvar|K}} or {{math|g}} on {{math|Γ}} are defined by setting the {{mvar|k}}-th successive minimum {{math|λ_k}} to be the infimum of the numbers {{math|λ}} such that {{math|λK}} contains {{mvar|k}} linearly-independent vectors of {{math|Γ}}. We have {{math|0 < λ₁ ≤ λ₂ ≤ ... ≤ λ_n < ∞}}.

The successive minima satisfyCassels (1957) p.156Cassels (1971) p.203Siegel (1989) p.57

$$\backslash frac\{2^n\}\{n!\}\; \backslash operatorname\{vol\}\backslash left(\backslash mathbb\{R\}^n/\backslash Gamma\backslash right)\; \backslash le\; \backslash lambda\_1\backslash lambda\_2\backslash cdots\backslash lambda\_n\; \backslash operatorname\{vol\}(K)\backslash le\; 2^n\; \backslash operatorname\{vol\}\backslash left(\backslash mathbb\{R\}^n/\backslash Gamma\backslash right).$$

A basis of linearly independent lattice vectors {{math| b₁, b₂, ..., b_n}} can be defined by {{math|1=g(b_j) = λ_j}}.

The lower bound is proved by considering the convex polytope {{math|2n}} with vertices at {{math|±b_j/ λ_j}}, which has an interior enclosed by {{mvar|K}} and a volume which is {{math|2ⁿ/n!λ₁ λ₂...λ_n}} times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by {{math|λ_j}} along each basis vector to obtain {{math|2ⁿ}} Simplex with lattice point vectors).

To prove the upper bound, consider functions {{math|f_j(x)}} sending points {{mvar|x}} in $K$ to the centroid of the subset of points in $K$ that can be written as $x\; +\; \backslash sum\_\{i=1\}^\{j-1\}\; a\_i\; b\_i$ for some real numbers $a\_i$. Then the coordinate transform $$x\text{'}\; =\; h(x)\; =\; \backslash sum\_\{i=1\}^\{n\}\; (\backslash lambda\_i\; -\backslash lambda\_\{i-1\})\; f\_i(x)/2$$ has a Jacobian determinant $J\; =\; \backslash lambda\_1\; \backslash lambda\_2\; \backslash ldots\; \backslash lambda\_n/2^n$. If $p$ and $q$ are in the interior of $K$ and $p-q\; =\; \backslash sum\_\{i=1\}^k\; a\_i\; b\_i$(with $a\_k\; \backslash neq\; 0$) then $$(h(p)\; -\; h(q))\; =\; \backslash sum\_\{i=0\}^k\; c\_i\; b\_i\; \backslash in\; \backslash lambda\_k\; K$$ with $c\_k\; =\; \backslash lambda\_k\; a\_k\; /2$, where the inclusion in $\backslash lambda\_k\; K$ (specifically the interior of $\backslash lambda\_k\; K$) is due to convexity and symmetry. But lattice points in the interior of $\backslash lambda\_k\; K$ are, by definition of $\backslash lambda\_k$, always expressible as a linear combination of $b\_1,\; b\_2,\; \backslash ldots\; b\_\{k-1\}$, so any two distinct points of $K\text{'}\; =\; h(K)\; =\; \backslash \{\; x\text{'}\; \backslash mid\; h(x)\; =\; x\text{'}\; \backslash \}$ cannot be separated by a lattice vector. Therefore, $K\text{'}$ must be enclosed in a primitive cell of the lattice (which has volume $\backslash operatorname\{vol\}(\backslash R^n/\backslash Gamma)$), and consequently $\backslash operatorname\{vol\}\; (K)/J\; =\; \backslash operatorname\{vol\}(K\text{'})\; \backslash le\; \backslash operatorname\{vol\}(\backslash R^n/\backslash Gamma)$.

{{Reflist}}

• {{cite book | first=J. W. S. | last=Cassels | author-link=J. W. S. Cassels | title=An introduction to Diophantine approximation | series=Cambridge Tracts in Mathematics and Mathematical Physics | volume=45 | publisher=Cambridge University Press | year=1957 | zbl=0077.04801 }}
• {{cite book | first=J. W. S. | last=Cassels | author-link=J. W. S. Cassels | title=An Introduction to the Geometry of Numbers | series=Classics in Mathematics | publisher=Springer-Verlag | edition=Reprint of 1971 | year=1997 | isbn=978-3-540-61788-4 }}
• {{cite book | first=Melvyn B. | last=Nathanson | title=Additive Number Theory: Inverse Problems and the Geometry of Sumsets | volume=165 | series=Graduate Texts in Mathematics | publisher=Springer-Verlag | year=1996 | isbn=0-387-94655-1 | zbl=0859.11003 | pages=180–185 }}
• {{cite book | last=Schmidt | first=Wolfgang M. | author-link=Wolfgang M. Schmidt | title=Diophantine approximations and Diophantine equations | series=Lecture Notes in Mathematics | volume=1467 | publisher=Springer-Verlag | year=1996 | edition=2nd | isbn=3-540-54058-X | zbl=0754.11020 | page=6 }}
• {{cite book | first=Carl Ludwig | last=Siegel | author-link=Carl Ludwig Siegel | title=Lectures on the Geometry of Numbers | publisher=Springer-Verlag | year=1989 | isbn=3-540-50629-2 | editor=Komaravolu S. Chandrasekharan | editor-link=Komaravolu S. Chandrasekharan | zbl=0691.10021 }}

Category:Hermann Minkowski

Category:Theorems in geometry