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Dual curvature measures and Minkowski problems Yang, Deane


In recent joint work with Károly Böröczky, Yong Huang, Erwin Lutwak, Deane Yang, Gaoyong Zhang, and Yiming Zhao, dual curvature measures of convex bodies in $\mathbb{R}^n$, which are conceptually dual to Federer's curvature measures, were constructed. This leads naturally to a Minkowski-type problem, which we call the dual Minkowski problem and which is equivalent to a Monge-Ampere PDE.
In particular, the dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\mathbb{R}^n$. A full solution to this when $1\lt q\lt n$ will be presented.
The necessary and sufficient condition is an explicit measure concentration condition. A variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral proved using a spherical partition. The second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body.

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