  We show that the only even, smooth, convex solutions to a class of isotropic
mixed Christoffel-Minkowski type problems are origin-centred spheres, which, in
particular, answers a question of Firey 74 in the even isotropic case about
kinematic measures. Employing the Heintze-Karcher inequality, we prove that the
only smooth, strictly convex solutions to a large class of Minkowski type
problems are origin-centred spheres. Immediate corollaries are the uniqueness
of solutions to the isotropic Orlicz-Minkowski problem and the isotropic
$L_p$-Gaussian-Minkowski problem when $p\geq 1$.
