On The Relative Distances Of Points In The Boundary Of A Plane Convex Body

Let C be a plane convex body.A chord pq of C is called an affine diameter of C,if there is no longer parallel chord in C.The ratio of |ab|to 1/2 |a1b1|,where a1b1 is an affine diameter of C parallel to ab,is called the C-distance of a and b,or the relative distance of a and b.We denote it by dC(a,b).Denote by φk(C)the greatest possible number d such that the boundary of C contains k points at pairwise distances at least d.Letφk(C)=sup{φk(C)|C∈c},where C denotes the family of plane convex bodies.In Chapter 1 we continue to focus on the pairwise relative distances of nine points in the boundary of a plane convex body.We try to determine the upper bound of the minimum pairwise relative distances of nine points in the boundary of a convex body,and it suffices to examine the minimum upper bound of the relative distances of every two consecutive vertices of convex nonagons.The main result of Chapter 1 is that every convex nonagon has two consecutive vertices at relative distance at most(?)-1,that is,φ9(c)≤(?)-1.According to the two configurations given by[14]and this paper,we get φ9(c)=(?)-1.In Chapter 2 we consider the pairwise relative distances of ten points in the boundary of a plane convex body.Z.Su,S.Li,J.Shen and L.Yuan disproved the conjecture that there exists no plane convex body whose boundary contains ten points at pairwise relative distances greater then 2/3 raised by Z.Langi,and they disproved the conjecture by giving a plane convex decagon whose pairwise relative distances of the vertices are greater than 2/3.In this paper we improve the lower bound for φ10(c)further to 17/25.The main result of this chapter is that there exists a plane convex decagon with the property that the relative distance of any two vertices is greater than 17/25,Lhat is φ10(c)>17/25>2/3.