Type: Article

Measure partitions using hyperplanes with fixed directions

Karasev, Roman N.; Roldán-Pensado, Edgardo; Soberón, Pablo;

Abstract:

This paper contributes to the theory of equally partitioning measures: Given t probability measures ?1,…,?t on Rd and a positive integer k, split Rd into certain subsets A1,…,Ak such that ?i(Aj)=1k for 1?i?t, 1?j?k. Most results concern measures that vanish on every hyperplane, and the sets Aj are finite unions of sets obtained by dissections of Rd by hyperplanes of prescribed directions. For example it is shown that, for t measures in the plane R2, there is a path formed by horizontal and vertical segments and using at most t?1 turns that splits R2 into two sets each having size 12 under all t measures.