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Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems. (English) Zbl 1372.52007

The Brunn-Minkowski theory of convex bodies, centering around mixed volumes and related notions, has in recent decades been extended in different directions. One of these variants is the dual Brunn-Minkowski theory, which, to be more precise, is not based on an exact duality but on what has been called ‘conceptual duality’. Therefore, the correct ‘dual’ notions are not always easy to find. The present paper succeeds with establishing a satisfactory dual of the classical curvature measures, which, however, have first to be transformed to the unit sphere, via the inverse radial mapping, before the ‘dual’ correspondence becomes a perfect formal analogy. In striking similarity to the classical case, the authors introduce the dual curvature measures and dual area measures via local Steiner-type formulas. The dual area measures turn out to be (spherical) integrals of powers of radial functions; the dual curvature measures are, heuristically speaking, image measures of these under a combination of radial and Gauss maps. Weak continuity and valuation properties of the dual curvature measures are proved. Special cases of the dual curvature measures are, somewhat surprisingly, the cone-volume measure and, up to a constant factor, Aleksandrov’s integral curvature of the polar body. It is shown that the dual curvature measures can be considered as differentials of the dual quermassintegrals. This is proved and expanded by establishing dual generalizations of Aleksandrov’s variational formula, involving Wulff shapes and their polars. This includes also a new proof of Aleksandrov’s classical variational lemma, without the use of inequalities for mixed volumes. The main result of the paper is then a Minkowski-type existence theorem, giving a sufficient condition on an even Borel measure on the unit sphere to be the \(k\)th dual curvature measure of an origin-symmetric convex body. The proof, which needs delicate estimates, is based on a maximization problem; the symmetry assumptions are required for the proof that it has a solution. As for cone-volume measures, the sufficient conditions include some subspace concentration property; the general necessity of these conditions is not treated in this paper.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
35J20 Variational methods for second-order elliptic equations
52A40 Inequalities and extremum problems involving convexity in convex geometry
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