Compactness of the L_p dual Minkowski problem in ℝ³

2025/05/22 by Boroczky, Karoly J., Chen, Shibing, Liu, Weiru +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.17219

Abstract

We prove the C⁰ estimate for the L_p qth dual Minkowski problem on S² under fairly general conditions; namely, when p lies in [0,1) and q>2+p, and the L_p qth dual curvarture is bounded and bounded away from zero. We note that it is known that the analogous C⁰ estimate does not hold if p<-1 and q=3. As a corollary of our C⁰ estimate, we deduce the uniqueness of the solution of the near isotropic qth L_p dual Minkowski problem on S² if q is close to 3 and the qth L_p dual curvature is Holder close to be the constant one function.

Citations

• Uniqueness in the near isotropic Lp dual Minkowski problem
• The L_p dual Christoffel-Minkowski problem for 1
• Dual Curvature Density Equation with Group Symmetry
• The L_p Gaussian Minkowski problem for C-pseudo-cones
• The dual Minkowski problem for positive indices
• Stability of the cone-volume measure with near constant density
• Uniqueness when the L_p curvature is close to be a constant for p∈[0,1)
• L^p-Minkowski Problem under Curvature Pinching
• Uniqueness of solutions to a class of isotropic curvature problems
• On the planar Gaussian-Minkowski problem
• The Logarithmic Minkowski conjecture and the L_p-Minkowski Problem
• Classification of solutions for the planar isotropic L_p dual Minkowski problem
• On the stability of the L_p-curvature
• Existence of non-symmetric solutions to the Gaussian Minkowski problem
• On a geometric combination of functions related to Prékopa-Leindler inequality
• The L_p-Minkowski problem with super-critical exponents
• The local logarithmic Brunn-Minkowski inequality for zonoids
• A non-existence result for the L_p-Minkowski problem
• Centro-Affine Differential Geometry and the Log-Minkowski Problem
• A sharp centro-affine isospectral inequality of Szegö--Weinberger type and the L^p-Minkowski problem
• The Minkowski problem in the Gaussian probability space
• On a non-homogeneous version of a problem of Firey
• On the local version of the Log-Brunn-Minkowski conjecture and some new\n related geometric inequalities
• Log-Brunn-Minkowski inequality under symmetry
• Non-uniqueness of solutions to the dual L_p-Minkowski problem
• On a generalised Blaschke-Santalò inequality
• Mass transportation functionals on the sphere with applications to the\n logarithmic Minkowski problem
• The L_p dual Minkowski problem for p>1 and q>0
• Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems
• Local L^p-Brunn-Minkowski inequalities for p < 1
• Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems
• Asymptotic behavior of flows by powers of the Gaussian curvature
• Flow by the power of the Gauss curvature
• On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities
• Remarks on the conjectured log-Brunn-Minkowski inequality
• Remarks on the conjectured log-Brunn–Minkowski inequality

Related

• The L_p dual Minkowski problem for p>1 and q>0
• Uniqueness and continuity of the solution to L_p dual Minkowski problem
• The L_p dual Minkowski problem for unbounded closed convex sets
• Diameter estimate for planar L_p dual Minkowski problem
• On the uniqueness of solutions to the isotropic L_p dual Minkowski problem