Name(s): Brian Kim High School(s): Stuyvesant High School, New York, NY

1. Name(s): Brian Kim High School(s): Stuyvesant High School, New York, NY Mentor: Dr. Dan Ismailescu Project Title: Packing and Covering with Centrally Symmetric Disks In this paper, we deal with arrangements of convex discs in the Euclidean plane. A convex disc is a plane compact convex set with nonempty interior. An arrangement of congruent copies of a convex disc K is a family A of convex discs, each of which is congruent to K. The arrangement is a packing if its members’ interiors are mutually disjoint, and it is a covering if the union of its members is the whole plane. An arrangement of translates of K by all vectors of a lattice is a lattice arrangement. Thus we may have lattice packings and lattice coverings of E with translates of K. The density of an arrangement A is, intuitively speaking, the ratio between the sum of the areas of members of A contained in an arbitrarily large region and the area of the region. Given a convex disc K, the lattice packing density of K, denoted by δL(K), is the maximum density of any lattice packing with congruent replicas of K. The analogous notion of lattice covering density, denoted by θL(K), is defined as the minimum density of any lattice covering with congruent copies of K. We prove that for every convex disc K we have that 1 ≤ δL(K)θ(K) ≤ 1.17225… The left inequality is tight and it improves a ten year old result.