  As we all know, the Minkowski type problem is the cornerstone of the
Brunn-Minkowski theory in Euclidean space. The Heisenberg group as a
sub-Riemannian space is the simplest non-Abelian degenerate Riemannian space
that is completely different from a Euclidean space. By analogy with the
Minkowski type problem in Euclidean space, the Minkowski type problem in
Heisenberg groups is still open. In the present paper, we develop for the first
time a sub-Riemannian version of Minkowski type problem in the horizontal
distributions of Heisenberg groups, and further give a positive answer to this
sub-Riemannian Minkowski type problem via the variational method.
