The dual Minkowski problem for negative indices

Abstract

Recently, the duals of Federer's curvature measures, called dual curvature measures, were discovered by Huang, Lutwak, Yang, and Zhang (ACTA, 2016). In the same paper, they posed the dual Minkowski problem, the characterization problem for dual curvature measures, and proved existence results when the index, q, is in (0,n). The dual Minkowski problem includes the Aleksandrov problem (q = 0) and the logarithmic Minkowski problem (q = n) as special cases. In the current work, a complete solution to the dual Minkowski problem whenever q < 0, including both existence and uniqueness, is presented.

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• Chord Measures in Integral Geometry and Their Minkowski Problems
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• The dual Minkowski problem for symmetric convex bodies
• Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems
• On the uniqueness of solutions to the isotropic $L_p$ dual Minkowski problem
• Non-uniqueness of solutions to the dual $L_p$-Minkowski problem
• On the Musielak-Orlicz-Gauss image problem
• The Gauss Image Problem with Weak Aleksandrov Condition
• The dual Orlicz-Minkowski problem
• $C^{1,1}$ regularity for solutions to the degenerate $L_p$ Dual Minkowski problem
• On a generalised Blaschke-Santalò inequality
• The $L_p$ Aleksandrov problem for origin-symmetric polytopes
• On the general dual Orlicz-Minkowski problem
• The Minkowski problem in the Gaussian probability space
• $L_p$-Steiner quermassintegrals
• The Minkowski problem based on the (p,q)-mixed quermassintegrals
• A flow method for the dual Orlicz-Minkowski problem
• Dual curvature measures on convex functions and associated Minkowski problems
• The general dual-polar Orlicz-Minkowski problem