Optimal constants in the Mahler–Weyl reduced-basis inequalities for lattices and symmetric convex bodies (Weyl 1942)

For a full lattice $\Lambda$ in $\mathbb{R}^n$ and a $0$-symmetric convex body $K$ with gauge function $f$, Weyl's 1942 paper 'On Geometry of Numbers' proved two basis inequalities. First (his Theorem V, found independently by Mahler): for every reduced basis $e_1,\ldots,e_n$ — one with $f(\sum_i x_ie_i)\ge f(e_k)$ whenever $\gcd(x_k,\ldots,x_n)=1$ — the normalized product $\mathrm{vol}(K)f(e_1)\cdots f(e_n)/\det(\Lambda)$ lies between $2^n/n!$ and $\mu_n=2^n(3/2)^{(n-1)(n-2)/2}$. Second (his Theorem VI): every pair $(\Lambda,K)$ admits some basis with the product at most $\rho_n$, an explicit constant of order $2^{\Theta(n\log n)}$. The two basis inequalities are due to Mahler (1938) and Weyl (1942) independently, and the determination of their exact constants is explicitly open in this lineage: Mahler (1964) asks for the exact value of the basis-existence constant at least for small $n$, and Remak (1938) records the true diagonal-product bound for Minkowski-reduced positive definite quadratic forms as unknown for larger $n$. This problem asks for the sharp constants $\mu_n$ and $\rho_n$ as functions of $n$, and for the sharp constant $c_n$ in $g_{11}\cdots g_{nn}\le c_nD$ for Minkowski-reduced forms (with $c_2=4/3$, $c_3=2$, $c_4=4$ sharp and classical, Mahler 1964 citing van der Waerden; this question is open from $n\ge 5$). The known interval for $\rho_n$ — from $2^n/n!\approx(2e/n)^n$ up to Weyl's value $\approx(n/e)^n$ — leaves even the exponential order of the optimal constants undetermined. The exact values of $\mu_n$ and $\rho_n$ are unknown in every dimension $n\ge 3$, and the exact value of $c_n$ is unknown for $n\ge 5$.

Background

A lattice $\Lambda$ in $\mathbb{R}^n$ is the set of all integer linear combinations of a basis of $\mathbb{R}^n$; its determinant $\det(\Lambda)$ is the volume of a fundamental cell, i.e. the volume of the parallelotope spanned by any of its bases. A $0$-symmetric convex body $K$ (a compact convex set with nonempty interior, symmetric under $x\mapsto -x$) determines a gauge function $f(x)=\inf\{t>0: x\in tK\}$, so that $K=\{x: f(x)\le 1\}$; conversely every norm on $\mathbb{R}^n$ arises this way. The successive minima $\lambda_1\le\cdots\le\lambda_n$ of $K$ with respect to $\Lambda$ are the smallest numbers $\lambda_k$ such that $\lambda_kK$ contains $k$ linearly independent points of $\Lambda$. Minkowski's second theorem, a cornerstone of the geometry of numbers, bounds their product:

$$\frac{2^n}{n!}\,\det(\Lambda)\;\le\;\mathrm{vol}(K)\,\lambda_1\cdots\lambda_n\;\le\;2^n\,\det(\Lambda),$$

and both inequalities are sharp (attained, for example, by the octahedron $\{x:\left|x_1\right|+\cdots+\left|x_n\right|\le 1\}$ with $\Lambda=\mathbb{Z}^n$ for the lower bound, and by the cube $[-1,1]^n$ with $\Lambda=\mathbb{Z}^n$ for the upper bound).

The vectors realizing the successive minima need not form a basis of $\Lambda$, which motivates basis-selection statements. An ordered basis $e_1,\ldots,e_n$ of $\Lambda$ is called reduced (in the sense of Minkowski, Mahler, and Weyl) if for every $k$ and all integers $x_1,\ldots,x_n$ with $\gcd(x_k,\ldots,x_n)=1$ one has $f(x_1e_1+\cdots+x_ne_n)\ge f(e_k)$; such bases always exist. For reduced bases the normalized products $\mathrm{vol}(K)f(e_1)\cdots f(e_n)/\det(\Lambda)$ fill a finite range: the First Finiteness Theorem, found independently by Mahler and Weyl, states

$$\frac{2^n}{n!}\;\le\;\frac{\mathrm{vol}(K)\,f(e_1)\cdots f(e_n)}{\det(\Lambda)}\;\le\;\mu_n, \qquad \mu_n=2^n\Big(\frac{3}{2}\Big)^{(n-1)(n-2)/2}.$$

The standard proofs lose a factor $3/2$ at each induction step, so $\mu_n$ grows like $2^{\Theta(n^2)}$ and is expected to be far from optimal, while the lower bound $2^n/n!\approx(2e/n)^n$ decreases rapidly in $n$. Weyl also proved a basis-existence version (his Theorem VI): every pair $(\Lambda,K)$ admits some basis $e_1,\ldots,e_n$ (not necessarily reduced) with $\mathrm{vol}(K)f(e_1)\cdots f(e_n)\le\rho_n\det(\Lambda)$, where $\rho_n=2^{2\nu-n}(2\nu+2)(2\nu+3)\cdots(\nu+n+1)$ and $\nu=\nu(n)$ is determined by $\nu^2-\nu/2-1\le n\le(\nu+1)^2-(\nu+1)/2-1$; thus $\rho_1,\ldots,\rho_7=2,4,12,42,168,756,3960$, and $\rho_n$ grows only like $2^{\Theta(n\log n)}$, roughly $(n/e)^n$ — dramatically faster than the lower bound but dramatically slower than $\mu_n$.

When $K$ is the unit ball of a positive definite quadratic form $Q$ (an ellipsoid; $f=\sqrt{Q}$), the reducedness condition above coincides with the classical Minkowski reduction of $Q$: writing $g_{ik}=Q(e_i,e_k)$ for the Gram matrix in the reduced basis and $D=\det(g_{ik})$ for the discriminant, Minkowski's reduction theory gives a bound $g_{11}\cdots g_{nn}\le c_nD$ for reduced forms, with $c_2=4/3$, $c_3=2$, $c_4=4$ sharp and classical (Mahler 1964, citing van der Waerden) and non-sharp reduction-theory values $c_5=10$, $c_6=42$, $c_7=250$; general upper bounds on $c_n$ follow from bounds on the Hermite constant $\gamma_n$ (the largest, over positive definite forms of determinant $1$ in $n$ variables, of the minimum of the form on nonzero integer vectors), notably those of Blichfeldt, refined by Remak. The exact $c_n$ is not known for $n\ge 5$.

Aitchison (1972) delimited the qualitative landscape: the finiteness properties underlying these theorems — that reducedness can be certified by finitely many inequalities, and that only finitely many unimodular transformations (changes of lattice basis by an integer matrix of determinant $\pm 1$) need be considered — hold for general bodies in dimensions at most $2$ but only for ellipsoids in dimensions $n\ge 3$; sharp general constants therefore cannot come from a finite case analysis. In modern computational lattice reduction, the quality of a basis is measured by the orthogonality defect, the ratio of the product of the basis vector lengths to the lattice determinant — exactly the Euclidean specialization of the products above; its optimal value for reduced bases is likewise unknown in general dimension. Recent adjacent results — a reverse Minkowski theorem bounding lattice-point counts from determinant constraints (Regev and Stephens-Davidowitz), and restricted successive minima over sublattices (Henk and Thiel) — address different quantities and do not determine Weyl's constants.

Problem Statement

Determine the sharp constants in the three inequalities (A)–(C) below, as functions of the dimension $n$.

(A) $\mu_n^{\mathrm{opt}}$: the smallest constant such that for every full lattice $\Lambda\subset\mathbb{R}^n$, every $0$-symmetric convex body $K$ with gauge $f$, and every reduced basis $e_1,\ldots,e_n$ of $\Lambda$,

$$\mathrm{vol}(K)\,f(e_1)\cdots f(e_n)\;\le\;\mu_n^{\mathrm{opt}}\,\det(\Lambda),$$

where a basis $e_1,\ldots,e_n$ of $\Lambda$ is called reduced (with respect to the gauge $f$) if for every $k=1,\ldots,n$ and all integers $x_1,\ldots,x_n$ with $\gcd(x_k,\ldots,x_n)=1$ one has $f(x_1e_1+\cdots+x_ne_n)\ge f(e_k)$ (such bases exist for every pair $(\Lambda,K)$). Known: $2^n/n!\le\mu_n^{\mathrm{opt}}\le 2^n(3/2)^{(n-1)(n-2)/2}$.

(B) $\rho_n^{\mathrm{opt}}$: the smallest constant such that every pair $(\Lambda,K)$ as above admits some basis $e_1,\ldots,e_n$ (not necessarily reduced) with

$$\mathrm{vol}(K)\,f(e_1)\cdots f(e_n)\;\le\;\rho_n^{\mathrm{opt}}\,\det(\Lambda).$$

Known: $2^n/n!\le\rho_n^{\mathrm{opt}}\le 2^{2\nu-n}(2\nu+2)(2\nu+3)\cdots(\nu+n+1)$, where $\nu=\nu(n)$ is the integer determined by $\nu^2-\nu/2-1\le n\le(\nu+1)^2-(\nu+1)/2-1$ (this upper bound is Weyl's explicit value; $\rho_1,\ldots,\rho_7=2,4,12,42,168,756,3960$).

(C) $c_n^{\mathrm{opt}}$: the smallest constant such that every Minkowski-reduced positive definite quadratic form $Q$ in $n$ variables with Gram matrix $(g_{ik})$ and discriminant $D=\det(g_{ik})$ satisfies

$$g_{11}\cdots g_{nn}\;\le\;c_n^{\mathrm{opt}}\,D,$$

where $Q$ is Minkowski-reduced when $(g_{ik})$ is its Gram matrix with respect to a basis that is reduced, in the sense of (A), for the gauge $f=\sqrt{Q}$. Known: $c_2=4/3$, $c_3=2$, $c_4=4$ are sharp and classical (Mahler 1964, citing van der Waerden); reduction theory gives the non-sharp values $c_5=10$, $c_6=42$, $c_7=250$, and general upper bounds via the Hermite constant (Minkowski, Blichfeldt, Remak), none sharp; the question is open from $n\ge 5$.

The three questions form one objective — the exact strength of the reduced-basis mechanism — and are related: a reduced basis is in particular a basis, so $\rho_n^{\mathrm{opt}}\le\mu_n^{\mathrm{opt}}$; and (C) is exactly (A) restricted to ellipsoid gauges, since the Minkowski-reduction condition on a form coincides with reducedness for the gauge $\sqrt{Q}$ (the constants then determine each other through the unit-ball volume). The exact value of each optimal constant is determined, as a function of $n$, as the corresponding supremum or infimum, and, where the extremum is attained, the extremal bodies, lattices, and bases are to be identified as well. The quantifiers in (A)–(C) range over all full lattices $\Lambda\subset\mathbb{R}^n$, all $0$-symmetric convex bodies $K$ (not only ellipsoids or polytopes), and the corresponding bases — reduced in (A) and (C), arbitrary in (B).

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