New affine inequalities and projection mean ellipsoids

  • Du Zou
  • Ge XiongEmail author


A variational formula for the Lutwak affine surface areas \(\Lambda _{j}\) of convex bodies in \(\mathbb {R}^n\) is established when \(1\le j\le n-1.\) By using introduced new ellipsoids associated with projection functions of convex bodies, we prove a sharp isoperimetric inequality for \(\Lambda _{j}\), which opens up a new passage to attack the longstanding Lutwak conjecture in convex geometry.

Mathematics Subject Classification

52A40 46B06 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsWuhan University of Science and TechnologyWuhanPeople’s Republic of China
  2. 2.School of Mathematical SciencesTongji UniversityShanghaiPeople’s Republic of China

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