The discrete logarithmic Minkowski problem
  Existence of solution of the logarithmic Minkowski problem is proved for the case where the discrete measures on the unit sphere satisfy the subspace concentration condition with respect to some special proper subspaces. In order to understand how optimal this condition is, we discuss certain conditions that any cone volume measure satisfies. 
Karoly J. Boroczky, Pal Hegedus, Guangxian Zhu
math.MG