There are several problems which can be called "Minkowski problem". Perhaps the most famous one is this problem from differential geometry:
"G. Minkowski proved an existence theorem for a closed convex surface with the Gaussian curvature given as a function of a unit normal under some natural condition on this function; the open question was: if the function is regular on a sphere, is the surface regular itself?
This is the citation from the Wikipedia article on Alexei Pogorelov, who made a major contribution to its solution Minkowski Problem.