On the Musielak-Orlicz-Gauss image problem

Abstract

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 Musielak-Orlicz-Gauss image measure, denoted by ${\tilde{C}}_{\Theta }(K,\cdot )$, involves a triple $\Theta =(G,\Psi ,\lambda )$ where $G$ and $\Psi$ are two Musielak-Orlicz functions defined on $S^{n-1}\times (0,\infty )$ 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 ${\tilde{V}}_{G,\lambda }(K)$ (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.

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Cited by in corpus (1)

• A flow approach to the Musielak-Orlicz-Gauss image problem