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Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems (2502.05436v1)
Published 8 Feb 2025 in math.MG
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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