On the general dual Orlicz-Minkowski problem

Abstract

For $K\subseteq {\mathbb{R}}^n$ a convex body with the origin $o$ in its interior, and $\varphi :{\mathbb{R}}^n\setminus \{o\}\to (0,\infty )$ a continuous function, define the general dual ($L_{\varphi })$ Orlicz quermassintegral of $K$ by $${\mathcal{V}}_{\varphi }(K)={\int }_{{\mathbb{R}}^n\setminus K}\varphi (x)dx.$$ Under certain conditions on $\varphi$, we prove a variational formula for the general dual ($L_{\varphi })$ Orlicz quermassintegral, which motivates the definition of ${\tilde{C}}_{\varphi ,\mathcal{V}}(K,\cdot )$, the general dual ($L_{\varphi })$ Orlicz curvature measure of $K$. We pose the following general dual Orlicz-Minkowski problem: {\it Given a nonzero finite Borel measure $\mu$ defined on $S^{n-1}$ and a continuous function $\varphi :{\mathbb{R}}^n\setminus \{o\}\to (0,\infty )$, can one find a constant $\tau >0$ and a convex body $K$ (ideally, containing $o$ in its interior), such that,} $$\mu =\tau {\tilde{C}}_{\varphi ,\mathcal{V}}(K,\cdot )?$$ Based on the method of Lagrange multipliers and the established variational formula for the general dual ($L_{\varphi })$ Orlicz quermassintegral, a solution to the general dual Orlicz-Minkowski problem is provided. In some special cases, the uniqueness of solutions is proved and the solution for $\mu$ being a discrete measure is characterized.

This paper has been accepted by Indiana University Mathematics Journal