Given a measure on the unit sphere, the classical Minkowski problem in convex geometry asks for necessary and sufficient conditions in order to construct a convex body in the Euclidean space whose surface area measure (or Gauss curvature when the measure has a density) is equal to the given measure. The partial differential equation associated with the Minkowski problem is a Monge-Ampere equation with measure data. We discuss unsolved major Minkowski problems for geometric measures in convex geometry, and mention recent breakthroughs in solving Minkowski problems via measure concentration conditions.

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