On Partitioning Colored Points

Let us imagine that red candles and blue candles are placed on the top of a cake. We want to cut it with a knife into two pieces in such a way that all the red candles are on one of the two pieces and all the blue candles are on the other piece. A question is when we can cut it successfully. The following Kirchberger’s theorem [5] answers this question in a general setting. Let X be a finite subset of Rd consisting of red points and blue points. We say that a subset S of X can be separated along the colors if there is a hyperplane h such that all the red points in S are contained in h and all the blue points in S are contained in h−. Here h and h− are the two open halfspaces associated with h. Kirchberger’s theorem states that if every d + 2 or fewer points in X can be separated along the colors, then all the points in X can be separated along the colors. We consider a more colorful cake cutting problem. Suppose that we have a cake with candles each of which is painted with one of k colors. We want to cut it with a knife by several times in such a way that all candles with the same color are on one of the pieces, although candles with different colors must not be on the same piece. Let us formally describe this setting. Let X be a finite subset of R d, and suppose that each point in X is painted with one of k colors. We say that a subset S of X can be partitioned by hyperplanes along the colors if there is a family F of hyperplanes satisfying the following three conditions: