The Study On The Interior Disjoint Empty Convex Partition Problem Of Finite Planar Point Sets

Let P denote a point set in general position in the plane,that is to say,no three points in P are collinear,H is a subset of P,Ch(H)is the convex hull of H,V(H)is the vertex set of H,I(H)is the set of points in P contained in Ch(H),which is called the inner point set for short.Let P1,P2,…,pk be the k(k?n)points of set P,where P1,P2,…,pk is marked consecutively.Let(P1,P2,…,Pk)denote a convex closed region.If the closed region(P1,P2,…,pk)is empty,let(P1P2…Pk)k denote an k-hole.For any two holes:S and T,if the intersection of Ch(S)and Ch(T)is empty,which the holes are with pairwise disjoint;if Ch(S)and Ch(T)have disjoint interiors,then we call the holes are with disjoint interiors.In this paper,we study the counting function m(kl,k2,…,kn),that is,to determine the minimum positive integer m(k1,k2,…,kn)(k1?k2?…?kn),such that any set with at least m(k1,k2,…,kn)points in the plane in general position contains a ki-hole for every i(1?i?n),in which the holes are with pairwise disjoint interiors.This paper mainly studies the counting function m(k1,…,ki)(2?i?4).We first divide the finite point set into several subsets,then discuss the subsets,and study the properties of subsets.On the basis of the existing research results,using the theory and analysis methods of convex sets,combined with geometric configurations,we obtain some new research results,which further enrich this research direction.The conclusions are as follows:the exact value of m(4,4,5)=11 is given,that is,any point set with at least 11 points in general position in the plane contains two 4-holes and one 5-hole with pairwise disjoint interiors;the range of 12?m(4,4,4,5)?13 is given,that is,12 or 13 is the minimum number,such that any point set with 12 or 13 points in general position in the plane contains three 4-holes and one 5-hole with pairwise disjoint interiors;the upper bound of m(5,5)?17 is deduced,that is,any point set with at least 17 points in general position in the plane contains two 5-holes with disjoint interiors;finally,the exact value of m(4,4,4,4)=11 is solved,that is,any point set with at least 11 points in general position in the plane contains four 4-holes with pairwise disjoint interiors.