Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 7351861, 5 pages

http://dx.doi.org/10.1155/2016/7351861

## The Projections of Convex Lattice Sets of Points in

College of Science, Beijing Forestry University, Beijing 100083, China

Received 10 May 2016; Revised 17 October 2016; Accepted 25 October 2016

Academic Editor: Francesco Soldovieri

Copyright © 2016 Yu Gu and Lin Si. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Can one determine a centrally symmetric lattice polygon by its projections? In 2005, Gardner et al. proposed the above discrete version of Aleksandrov’s projection theorem. In this paper, we define a coordinate matrix for a centrally symmetric convex lattice set and suggest an algorithm to study this problem.

#### 1. Introduction

According to Gardner [1], geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross sections by planes. When the geometric object is a convex body, there are many results from convex geometry. A famous example is the celebrated Aleksandrov’s projection theorem, that is, for an origin symmetric convex body (compact, convex subset with nonempty interior in the -dimensional Euclidean space ); it can be determined by its projections. For more details, see [1]. When the object is a convex lattice set, it models the atoms in a crystal which will lead to a range of topics concerned with electron microscopy in discrete tomography [2, 3].

If is a set in , we denote the convex hull of by . The dimension of is the dimension of its affine hull and is denoted by . The notation for the usual orthogonal projection of on a subspace is . A convex lattice set is a finite subset of the -dimensional integer lattice such that and we denote the cardinality of by . Let denote the set of all centrally symmetric convex lattice sets about origin of .

In 2005, Gardner et al. [4] proposed the following discrete version of Aleksandrov’s projection theorem.

*Question 1. *Let and belong to with . If holds for every hyperplane with a norm , do and coincide up to a translation?

In [4], Gardner et al. discovered a counterexample to this problem in with cardinality (Figure 1). In [5], Zhou proved that the counterexample discovered by Gardner et al. was the only counterexample in , up to unimodular transformations and with cardinality not larger than 17. In [6], Xiong proved that an origin symmetric convex lattice set can be uniquely determined by its lattice projection counts if -coordinate’s absolute value is not bigger than 2 for every point in this origin symmetric convex lattice set and its cardinality is not 11.