Lattice Delone simplices with super-exponential volume

In this short note we give a construction of an infinite series of Delone simplices whose relative volume grows super-exponentially with their dimension. This dramatically improves the previous best lower bound, which was linear. BACKGROUND Consider the Euclidean space Rd with norm ‖ · ‖ and a discrete subset Λ ⊂ Rd. A d-dimensional polytope L = conv{v0, . . . ,vn} with v0, . . . ,vn ∈ Λ is called a Delone polytope of Λ, if there exists an empty sphere S with S∩Λ = {v0, . . . ,vn}. That is, if there is a center c ∈ Rd and a radius r > 0 such that ‖vi − c‖ = r for i = 0, . . . , n, and ‖v − c‖ > r for the remaining v ∈ Λ \ {v0, . . . ,vn}. If the Delone polytope is a simplex, hence n = d, we speak of a Delone simplex. Over the past decades Delone polytopes experienced a renaissance in applications like computer graphics and computational geometry, where they are traditionally called Delaunay polytopes due to the French transcription of Delone. Historically, Delone polytopes received attention in the study of positive definite quadratic forms (PQFs), in particular in a reduction theory due to Voronoi (cf. [Vor08]). To every d-ary PQF Q, a point lattice Λ = AZd with Q = AtA is associated; Λ is a discrete set and uniquely determined by Q up to orthogonal transformations. A d-dimensional polytope L = conv{v0, . . . ,vn}, with v0, . . . ,vn ∈ Zd, is a Delone polytope of the lattice Λ (and also called a Delone polytope of Q), if and only if there exists a c ∈ Rd and r > 0 with Q[v − c] = ‖A(v − c)‖2 ≥ r2 for all v ∈ Zd and with equality if and only if v = vi for i ∈ {0, . . . , n}. The set of all Delone polytopes forms a periodic face-to-face tiling of Rd. It is called the Delone subdivision of Q. If all Delone polytopes are simplices we speak of a Delone triangulation. Delone subdivisions form a poset with respect to refinement, in which triangulations are maximally refined elements. Two Delone polytopes L and L′ are unimodularly equivalent, if L = UL′ + t for some unimodular transformation U ∈ GLd(Z) and a translation vector t ∈ The first author was partially supported by the Spanish Ministry of Science and Education under grant MTM2005-08618-C02-02. The second and third author were partially supported by the “Deutsche Forschungsgemeinschaft” (DFG) under grant SCHU 1503/4-1. The third author was partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). 1 2 FRANCISCO SANTOS, ACHILL SCHÜRMANN, AND FRANK VALLENTIN Z d. Voronoi [Vor08] showed that, up to unimodular equivalence, there exist only finitely many Delone subdivisions in each dimension d. Delone simplices may be considered as “building blocks” in this theory and therefore their classification is of particular interest. For more information on the classical theory we refer the interested reader to [SV05] or the original works of Voronoi [Vor08] and Delone [Del37]. BOUNDS FOR RELATIVE VOLUME Let Λ ⊂ Rd be a lattice. A lattice simplex conv{v0, . . . ,vn} is called unimodular if {v0, . . . ,vn} is an affine basis of Λ. All unimodular simplices are unimodularly equivalent and, in particular, have the same volume. The relative volume (or normalized volume) of a lattice simplex L is the volume of L divided by that of a unimodular simplex of Λ, so that the relative volume equals vol(L) · d!. Equivalently, it equals the index in Λ of the sublattice affinely spanned by the vertices of L. Clearly, the relative volume is an invariant with respect to unimodular transformations. Hence, in order to classify possible Delone simplices up to unimodular equivalence, a first question, already raised by Delone in [Del37], is what is the maximum relative volume mv(d) of d-dimensional Delone simplices. The sequence mv(d) is (weakly) increasing, since from a Delone simplex for a lattice Λ ⊂ Rd one can easily construct another of the same relative volume for the lattice Λ× Z ⊂ Rd+1. Voronoi knew that up to dimension d = 4 all Delone simplices have relative volume 1, while there are Delone simplices of volume 2 in d = 5. In [Bar73] Baranovskii proved mv(5) = 2, and later Baranovskii and Ryshkov [BR98] proved mv(6) = 3. Dutour classified all 6-dimensional Delone polytopes in [Dut04]. Ryshkov [Rys76] was the first who proved that relative volumes of Delone simplices are not bounded when the dimension goes to infinity. More precisely, for every k ∈ N he constructed Delone simplices of relative volume k in dimension 2k + 1, establishing that mv(d) ≥ ⌊ d−1 2 ⌋ . This was recently improved to the still linear lower bound mv(d) ≥ d− 3 by Erdahl and Rybnikov [ER02]. In this note we prove the following two lower bounds on mv(d): Theorem 1. For every pair of dimensions d1 and d2, mv(d1 + d2) ≥ mv(d1)mv(d2). Theorem 2. For every dimension of the form d = 2n − 1, mv(d) ≥ (d+ 1)/4. Theorem 1 immediately implies exponential lower bounds on mv(d). For example, mv(5) = 2 gives mv(d) ≥ 2⌊d/5⌋ ∼ 1.1487d. Even better, it is known that mv(24) ≥ 20480. Indeed, the Delone subdivision of the Leech lattice, which was determined by Borcherds, Conway, Queen, Parker and Sloane [CS88, Chapter 25], contains simplices of relative volume 20480 (the simplex denoted a24 1 a1 in their classification). Therefore we obtain: LATTICE DELONE SIMPLICES WITH SUPER-EXPONENTIAL VOLUME 3 Corollary 1. mv(d) ≥ 20480 ∼ 1.5123. Theorem 2 gives a much better lower bound asymptotically: Corollary 2. log(mv(d)) ∈ Θ(d log d). Proof. For the lower bound, let 2n be the largest power of two that is smaller or equal to d+1 (so that 2n ≥ (d+1)/2). Theorem 2, together with the monotonicity of mv(d), gives: mv(d) ≥ mv(2 − 1) ≥ (2 n)(2 +2)/2 42n−1 ∈ 2Θ(n2n) = 2 log . For the upper bound we use the following argument, which is Lemma 14.2.5 in [DL97] (attributed to L. Lovasz): Given a Delone simplex L of some PQF, the volume of the centrally symmetric difference body L− L = {v − v : v,v′ ∈ L} is vol(L − L) = (2d d ) vol(L) (see [RS57]). The polytope L − L does not contain elements of L\{0} in its interior. Thus, by Minkowski’s fundamental theorem (see [GL87], §5 Theorem 1) we know that vol(L−L) ≤ 2d. Putting things together we get mv(d) ≤ 2 dd! (2d d ) ∼ √ 2πd (