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小区x_2小区…小区x_g of the plane into g disjoint convex polygons suchevery x_i contains exactly a red points and b blue points. This theorem was proved by a . Kanoand M.Kano for a=1,2 and they proposed it as a conjecture，and then this conjecture was independently proved by三个papers.We obtain the following three newresults.(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 + 2毫克,then we can subdivide the plane into g_1+g_2 disjoint convex polygons x_1小区…x_ {g_1}小区…小区{g_2}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小区…小区x_g小区y_1…小区y_h of the plane into g+h disjoint convexpolygons such that every x_i contains exactly a red points and a+1 blue points and every y_jcontains exactly a+1 red points and a blue points.(3)If|R|=a(g_1+g_2)+(a+1)g_3and | b |= bg 1 +(b + 1)(g译文:2 + 3 g _),then there exists a subdivision x_1小区…小区x_ {g_1}小区y_1小区…小区y_ {g_2}小区z_{1}小区…小区z_ {g_3} of the plane intog_1+g_2+g_3 disjoint convex polygons such that every x_i contains exactly a red points and b bluepoints, every y_i, if any, contains exactly a red points and b+1 blue points, and every z_i，if any,a+1 red points and b+1 blue pointsWe also study some problems on discrete geometry与graph theory相关的问题。

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.