Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems

Abstract

A longstanding question in the dual Brunn-Minkowski theory is what are the dual analogues of Federer's curvature measures for convex bodies. The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems, which answer the question of what are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body. Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.

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• A flow approach to the Musielak-Orlicz-Gauss image problem
• The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures
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• The Hadwiger theorem on convex functions, I
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• Valuations on Log-Concave Functions
• A flow method for a generalization of $L_p$ Christofell-Minkowski problem
• Affine isoperimetric inequalities on flag manifolds
• Non-uniqueness of solutions to the dual $L_p$-Minkowski problem
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• The (Self-Similar, Variational) Rolling Stones
• Toward a relative q-entropy
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• Flow by Gauss curvature to Dual Orlicz-Minkowski problems
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• The $L_p$ Aleksandrov problem for origin-symmetric polytopes
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• A flow method for the dual Orlicz-Minkowski problem
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