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Abstract To the families of geometric measures of convex bodies (the area measures of Aleksandrov‐Fenchel‐Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their logarithmic variants are proposed and attacked. When the given ‘data’ is sufficiently regular, these problems are a new type of fully nonlinear partial differential equations involving dual quermassintegrals of functions. Major cases of these Minkowski problems are solved without regularity assumptions.
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This content will become publicly available on July 1, 2026
Minkowski problems for geometric measures
This paper describes the theory of Minkowski problems for geometric measures in convex geometric analysis. The theory goes back to Minkowski and Aleksandrov and has been developed extensively in recent years. The paper surveys classical and new Minkowski problems studied in convex geometry, PDEs, and harmonic analysis, and structured in a conceptual framework of the Brunn-Minkowski theory, its extensions, and related subjects.
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- Award ID(s):
- 2005875
- PAR ID:
- 10633777
- Publisher / Repository:
- American Mathematical Society
- Date Published:
- Journal Name:
- Bulletin of the American Mathematical Society
- Volume:
- 62
- Issue:
- 3
- ISSN:
- 0273-0979
- Page Range / eLocation ID:
- 359 to 425
- Subject(s) / Keyword(s):
- Convex body polytope Minkowski problem logarithmic Minkowski problem dual Minkowski problem Lp Minkowski problem quermassintegral dual quermassintegral chord integral surface area measure cone-volume measure area measure curvature measure dual curvature measure chord measure
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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