The dual Minkowski problem for symmetric convex bodies

Károly J. Böröczky, Erwin Lutwak, Deane Yang, Gaoyong Zhang, Yiming Zhao

Research output: Contribution to journalArticlepeer-review


The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on a prescribed even measure on the unit sphere for it to be the q-th dual curvature measure of an origin-symmetric convex body in Rn. A full solution to this is given when 1<q<n. The necessary and sufficient conditions turn out to be an explicit measure concentration condition. To obtain the results, 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 which is obtained by using a spherical partition. The second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body.

Original languageEnglish (US)
Article number106805
JournalAdvances in Mathematics
StatePublished - Nov 7 2019


  • Convex body
  • Dual Minkowski problem
  • Minkowski problem
  • Radial Gauss image
  • Subspace concentration condition

ASJC Scopus subject areas

  • General Mathematics


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