On Generosity And Related Problems

Let gL be a family consisting of all sets congruent with L in Rd,and K(?)gL be a compact set.If for any two different points x,y?K,there exists a set L'?E gL such that x,y?L'and L'(?)K,then the set K is called gL-convex.We also say that L is generous(to K),and K is grateful(to L).Let FL be a family consisting of all sets similar to L in Rd.If.FL contains all compact FL-convex sets,then L is called selfish.The thesis mainly studies the generosity and gratefulness of discrete point sets.In chapter 1 we first study generosity of non-selfish sets,and get the vertex set of an isosceles triangle is generous in R2 if and only if its apex angle ? ?{?/5,?/2,3?/5,2?/3}.Then we obtain some general properties of generous discrete point sets.In chapter 2 we investigate the gratefulness of finite point sets in R2,and the vertex sets of platonic solids and Archimedean solids.We prove that the vertex sets of Archimedes solids are grateful,and any subset L of an Archimedean solid K is generous if and only if L can realize all the distances in K.In addition,we estimate the grateful strength of each object.In chapter 3,we study the gratefulness of the k-skeleton of platonic solids and prove that all k-skeletons(k=1,2,3)of the platonic solids are grateful.Moreover,we probe into the relation among the gratefulness of the various skeletons.In chapter 4,we obtain that the circular cylinder is grateful,which further supplies the convex body of gratefulness.