On the general dual Orlicz-Minkowski problem

For $K\subseteq \mathbb{R}^n$ a convex body with the origin $o$ in its interior, and $ϕ:\mathbb{R}^n\setminus\{o\}\rightarrow(0, \infty)$ a continuous function, define the general dual ($L_ϕ)$ Orlicz quermassintegral of $K$ by $$\mathcal{V}_ϕ(K)=\int_{\mathbb{R}^n \setminus K} ϕ(x)\,dx.$$ Under certain conditions on $ϕ$, we prove a variational formula for the general dual ($L_ϕ)$ Orlicz quermassintegral, which motivates the definition of $\widetilde{C}_{ϕ,\mathcal{V}}(K, \cdot)$, the general dual ($L_ϕ)$ Orlicz curvature measure of $K$. We pose the following general dual Orlicz-Minkowski problem: {\it Given a nonzero finite Borel measure $μ$ defined on $S^{n-1}$ and a continuous function $ϕ: \mathbb{R}^n\setminus\{o\}\rightarrow (0, \infty)$, can one find a constant $τ>0$ and a convex body $K$ (ideally, containing $o$ in its interior), such that,} $$μ=τ\widetilde{C}_{ϕ,\mathcal{V}}(K,\cdot)? $$ Based on the method of Lagrange multipliers and the established variational formula for the general dual ($L_ϕ)$ 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 $μ$ being a discrete measure is characterized.

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