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We prove the following generalization of the ham sandwich theorem, conjectured by Imre Bárány. Given a positive integer  k  and  d  nice measures  μ  1 , μ  2 ,…, μ   d   in ℝ  d   such that  μ   i  (ℝ  d  )= k  for all  i , there is a partition of ℝ  d   into  k  interior‐disjoint convex parts  C  1 , C  2 ,…, C   k   such that  μ   i   ( C   j  )=1 for all  i , j . If  k =2, this gives the ham sandwich theorem. This result was proved independently by R. N. Karasev.

BALANCED CONVEX PARTITIONS OF MEASURES IN ℝ d