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2 results

  • DUAL $L_{p}$ JOHN ELLIPSOIDS

  • Wuyang Yu, Gangsong Leng, Donghua Wu
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 50 / Issue 3 / October 2007
    Published online by Cambridge University Press:
    08 January 2008, pp. 737-753
    • Article
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  • In this paper, the dual $L_p$ John ellipsoids, which include the classical Löwner ellipsoid and the Legendre ellipsoid, are studied. The dual $L_p$ versions of John's inclusion and Ball's volume-ratio inequality are shown. This insight allows for a unified view of some basic results in convex geometry and reveals further the amazing duality between Brunn–Minkowski theory and dual Brunn–Minkowski theory.

  • $L_p$ John Ellipsoids

  • Erwin Lutwak, Deane Yang, Gaoyong Zhang
  • Journal:
    Proceedings of the London Mathematical Society / Volume 90 / Issue 2 / March 2005
    Published online by Cambridge University Press:
    25 February 2005, pp. 497-520
    Print publication:
    March 2005

  • It is shown that the classical John ellipsoid, the Petty ellipsoid and a recently discovered ‘dual’ of the Legendre ellipsoid are all special cases ($p = \infty$, $p = 1$ and $p = 2$) of a family of $L_p$ ellipsoids which can be associated with a fixed convex body. This insight allows for a unified view of, alternative approaches to, and extensions of some basic results in convex geometry.