On Arnold's Problem on the Classifications of Convex Lattice Polytopes

In 1980, V.I. Arnold studied the classification problem for convex lattice polygons of given area. Since then this problem and its analogues have been studied by B'ar'any, Pach, Vershik, Liu, Zong and others. Upper bounds for the numbers of non-equivalent ddimensional convex lattice polytopes of given volume or cardinality have been achieved. In this paper, by introducing and studying the unimodular groups acting on convex lattice polytopes, we obtain lower bounds for the number of non-equivalent d-dimensional convex lattice polytopes of bounded volume or given cardinality, which are essentially tight.Comment: 15 pages, 3 figure

Minkowski Bisectors, Minkowski Cells, and Lattice Coverings

This article introduces and studies Minkowski Bisectors, Minkowski Cells, and Lattice Coverings.Comment: 9 figure

On the translative packing densities of tetrahedra and cubooctahedra

In this paper, upper bounds for the densities of the densest translative tetrahedron packings and the densest translative cubooctahedron packings are obtained.Comment: 37 pages, 9 figure

Characterization of the Two-Dimensional Six-Fold Lattice Tiles

This paper characterizes all the convex domains which can form six-fold lattice tilings of the Euclidean plane. They are parallelograms, centrally symmetric hexagons, one type of centrally symmetric octagons and two types of decagons.Comment: 20 pages, 14 figures. arXiv admin note: substantial text overlap with arXiv:1712.0112

A Mathematical Theory for Random Solid Packings

Packings of identical objects have fascinated both scientists and laymen alike for centuries, in particular the sphere packings and the packings of identical regular tetrahedra. Mathematicians have tried for centuries to determine the densest packings; Crystallographers and chemists have been fascinated by the lattice packings for centuries as well. On the other hand, physicists, geologists, material scientists and engineers have been challenged by the mysterious random packings for decades. Experiments have shown the existence of the dense random sphere packings and the loose random sphere packings for more than half a century. However, a rigorous definition for them is still missing. The purpose of this paper is to review the random solid packings and to create a mathematical theory for it.Comment: 6 pages, 1 figur

Classification of the sublattices of a lattice

In 1945-46, C. L. Siegel proved that an $n$-dimensional lattice $\Lambda$ of determinant ${\rm det}(\Lambda )$ has at most $m^{n^2}$ different sublattices of determinant $m\cdot {\rm det}(\Lambda )$. In 1997, the exact number of the different sublattices of index $m$ was determined by Baake. This paper presents a systematic treatment for counting the sublattices and deduces a formula for the number of the sublattice classes of determinant $m\cdot {\rm det}(\Lambda )$.Comment: 8 page

A quantitative program for Hadwiger's covering conjecture and Borsuk's partition conjecture

In this article we encode Hadwiger's covering conjecture and Borsuk's partition conjecture into continuous functions defined on the spaces of convex bodies, propose a four-step program to approach them, and obtain some partial results.Comment: 17 pages, four figure

Characterization of the Two-Dimensional Five-Fold Lattice Tiles

In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. It is known that there is no other convex domain which can form a two-, three- or four-fold lattice tiling in the Euclidean plane, but there is a centrally symmetric convex decagon which can form a five-fold lattice tiling. This paper characterizes all the convex domains which can form a five-fold lattice tiling of the Euclidean plane.Comment: 20 pages, 14 figures. arXiv admin note: text overlap with arXiv:1711.02514, arXiv:1710.0550

A note on lattice coverings

This note presents an interesting counterexample to a basic covering problem.Comment: 3 pages, 3 figure

Classification of Convex lattice polytopes

In this paper we study the classification problem of convex lattice ploytopes with respect to given volume or given cardinality