On the Musielak-Orlicz-Gauss Image Problem

In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of convex bodies. For a convex body K , its MusielakOrlicz-Gauss image measure, denoted by C (Theta) (K, <middle dot> ) , involves a triple Theta = (G, Psi , lambda) where G and Psi are two Musielak-Orlicz functions defined on S( n - 1 )x ( 0 , infinity) , and lambda is a nonzero finite Lebesgue measure on the unit sphere S (n -1) . Such a measure can be produced by a variational formula of V (G,lambda) (K) (the general dual volume of K with respect to lambda ) under the perturbations of K by the Musielak-Orlicz addition defined via the function Psi . The Musielak-Orlicz-Gauss image problem contains many intensively studied Minkowski-type problems and the recent Gauss image problem as its special cases. Under the condition that G is decreasing on its second variable, the existence of solutions to this problem is established.