2024/12/18 by Na Fu, J F Sun, Chen, Bin +2
Mathematics · #Point processes and geometric inequalities #Mathematical Approximation and Integration #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2412.13557
This article introduces the Lp-Gauss dual curvature measure and proposes its related Lp-Gauss dual Minkowski problem as: for p,q∈ℝ, under what necessary and/or sufficient condition on a non-zero finite Borel measure μ on unit sphere does there exist a convex body K such that μ is the Lp Gauss dual curvature measure? If K exists, to what extent is it unique? This problem amounts to solving a class of Monge-Ampère type equations on unit sphere in smooth case: e-(|∇ hK|2+hK2)/(2)hK1-p (|∇ hK|2+hK2)(q-n)/(2) det(∇2hK+hKI)=f, (0.1) where f is a given positive smooth function on unit sphere, hk is the support function of convex body K, ∇ hK and ∇2hK are the gradient and Hessian of hK on unit sphere with respect to an orthonormal basis, and I is the identity matrix. We confirm the existence of solution to the new problem with p,q>0 and the existence of smooth solution to the equation (0.1) with p ,q∈ℝ by variational method and Gaussian curvature flow method, respectively. Furthermore, the uniqueness of solution to the equation (0.1) in the case p,q∈ℝ with q