On Polyhedral Cages Holding Discs

For any subset M(?)Rd,if for any two points x,y∈M,the closed line segment between the two points is contained in M,then M is called a convex set.For any subset V of Rd,the smallest convex set containing V,namely,the intersection of all convex sets in Rd containing V is called the convex hull of V.A polytope is the convex hull of a non-empty finite set ｛x1,x2,…,xn}.A cage is the 1-skeleton of a 3-dimensional polytope in R3.A cage G is said to hold a convex body B if no rigid motion can bring B far away without meeting G on its way.In Chapter 1,we prove that there are tetrahedral cages holding exactly n unit discs,where n ∈ {0,1,2,3,4,6,8,12,16},and there is no such cage for any other n.In Chapter 2,we investigate the possible values n such that there is a pentahedral cage holding exactly n discs,and show that when n ∈[0,41]U {44,49,50,51,56,57} such cages exist.In Chapter 3 we discuss the number of discs held by a regular pyramidal cage over an n regular polygon,and prove that it can hold at most(2n2-n+1)discs,where n ≥ 4 and n is an even number.