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.

Original languageEnglish (US)
Pages (from-to)13-38
Number of pages26
JournalMilan Journal of Mathematics
Issue number1
StatePublished - Jun 2011

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Aleksandrov-Fenchel inequality
  • mixed volume
  • optimal transport map
  • quermassintegral


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