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On Aleksandrov-Fenchel Inequalities for k-Convex Domains

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In this lecture notes, we will discuss the classical Aleksandrov-Fenchel inequalities for quermassintegrals on convex domains and pose the problem of how to extend the inequalities to non-convex domains. We will survey some recent progress on the problem, then report some of our joint work [9] in which we generalize the k-th stage of the inequalities to a class of (k + 1)-convex domains. Our proof, following the earlier work of Castillion [8] for k =  1 case of the inequalities, uses the method of optimal transport.

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Correspondence to Sun-Yung Alice Chang.

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The research of the first author is partially supported by NSF grant DMS-0758601.

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Chang, SY.A., Wang, Y. On Aleksandrov-Fenchel Inequalities for k-Convex Domains. Milan J. Math. 79, 13–38 (2011). https://doi.org/10.1007/s00032-011-0159-2

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