Kadets type theorems for partitions of a convex body
  For convex partitions of a convex body $B$ we prove that we can put a homothetic copy of $B$ into each set of the partition so that the sum of homothety coefficients is $\ge 1$. In the plane the partition may be arbitrary, while in higher dimensions we need certain restrictions on the partition. 
Arseniy Akopyan and Roman Karasev
math.CO math.FA