On the General Dual Orlicz-Minkowski Problem

For K ⊆ ℝ n a convex body with the origin o in its interior, and φ : ℝ n \ {o} → (0,∞) a continuous function, define the general dual (L φ) Orlicz quermassintegral of K by V Φ ( K ) = ∫ ℝ n \ K Φ ( x ) d x . Under certain conditions on φ, we prove a variational formula for the general dual (Lφ) Orlicz quermassintegral, which motivates the definition of C̃ φ, V(K, ·), the general dual (Lφ) Orlicz curvature measure of K. We pose the following general dual Orlicz-Minkowski problem: Given a nonzero finite Borel measure μ defined on S n−1 and a continuous function φ : ℝ n \ {o} → (0,∞), can one find a constant τ > 0 and a convex body K (ideally, containing o in its interior), such that μ = τC̃φ, V (K, ·)? Based on the method of Lagrange multipliers and the established variational formula for the general dual (Lφ) 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 given.