BALANCED PARTITIONS OF TWO SETS OF POINTS IN THE PLANE

Recently，the Ham-sandwich Theorem was generalized as follows：If|R|=ag and|B|=bg，then there exists a subdivision X_1∪X_2∪...∪X_g of the plane into g disjoint convex polygons such that every X_i contains exactly a red points and b blue points.This theorem was proved by A.Kano and M.Kano for a=1，2 and they proposed it as a conjecture，and then this conjecture was independently proved by three papers.We obtain the following three new results.(1)Suppose that$R$is a disjoint union of R_1 and R_2.Let|R_1|=g_1and|R_2|=g_2.If|B|=(m-1)g_1+mg_2，Then we can subdivide the plane into g_1+g_2 disjoint convex polygons X_1∪···X_{g_1}∪Y_1∪···∪Y_{g_2}so that every X_i contains exactly one red point of R_1and m-1blue points，and every Y_j contains exactly one red point of R_2and m blue points.(2)Let a≥1，g≥0 and h≥0be integers such that g+h≥1.If|R|ag=(a+1)h and B|=(a+h)，Then there exists a subdivision X_1∪…∪X_g∪Y_1∪……∪Y_h of the plane into g+h disjoint convex polygons such that every X_i contains exactly a red points and a+1 blue points and every Y_j contains exactly a+1red points and a blue points.(3)If|R|=a(g_1+g_2)+(a+1)g_3 and|B|=bg_1+(b+1)(g_2+g_3)，Then there exists a subdivision X_1∪X_{g_1}∪Y_1∪Y_{g_2}∪Z_{1}∪Z_{g_3}of the plane into g_1+g_2+g_3 disjoint convex polygons such that every X_i contains exactly a red points and b blue points，every Y_i，if any，contains exactly a red points and b+1 blue points，and every Z_i，if any，contains exactly a red points and b+1 blue points，and every Z_i，if any，Contains exactly a+1red points and b+1blue pointsWe also study some problems on discrete geometry and graph theory related to the above problems。

Recently, the Ham-sandwich Theorem was generalized as follows : If|R|=ag and|B|=bg, then there exists a subdivision X_1∪X_2∪・・・∪X_g of the plane into g disjoint convex polygons such that every X_i contains exactly a red points and b blue points. This theorem was proved by A. Kano and M.Kano for a=1,2 and they proposed it as a conjecture, and then this conjecture was independently proved by three papers.We obtain the following three new results.(1)Suppose that $R$ is a disjoint union of R_1 and R_2. Let|R_1|=g_1 and|R_2|=g_2. If|B|=(m-1)g_1+ mg_2, then we can subdivide the plane into g_1+g_2 disjoint convex polygons X_1∪・・・X_{g_1}∪Y_1∪・・・∪Y_{g_2} so that every X_i contains exactly one red point of R_1 and m-1 blue points, and every Y_j contains exactly one red point of R_2 and m blue points.(2)Let a≧1, g≧0 and h≧0 be integers such that g+h≧1. If|R|=ag+(a+1)h and|B|=(a+1)g+ah, then there exists a subdivision X_1∪・・・∪X_g∪Y_1∪・・・∪ Y_h of the plane into g+h disjoint convex polygons such that every X_i contains exactly a red points and a+1 blue points and every Y_j contains exactly a+1 red points and a blue points.(3)If|R|=a(g_1+g_2)+(a+1)g_3 and|B|=bg_1+(b+1)(g_2+g_3), then there exists a subdivision X_1∪・・・∪X_{g_1}∪Y_1∪・・・∪ Y_{g_2}∪Z_{1}∪・・・∪Z_{g_3} of the plane into g_1+g_2+g_3 disjoint convex polygons such that every X_i contains exactly a red points and b blue points, every Y_i, if any, contains exactly a red points and b+1 blue points, and every Z_i, if any, contains exactly a+1 red points and b+1 blue pointsWe also study some problems on discrete geometry and graph theory related to the above problems.