Skip to main content
    • Aa
    • Aa

On the reverse Lp–busemann–petty centroid inequality

  • Stefano Campi (a1) and Paolo Gronchi (a2)

The volume of the Lp-centroid body of a convex body K ⊂ ℝd is a convex function of a time-like parameter when each chord of K parallel to a fixed direction moves with constant speed. This fact is used to study extrema of some affine invariant functionals involving the volume of the Lp-centroid body and related to classical open problems like the slicing problem. Some variants of the Lp-Busemann-Petty centroid inequality are established. The reverse form of these inequalities is proved in the two-dimensional case.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[CG] S. Campi and P. Gronchi , The Lp-Busemann-Petty centroid inequality body, Adv. Math. 167 (2002), 128141.

[FR] I. Fáry and L. Rédei . Der zentralsymmetrische Kern und die zentralsymmetrische Hüllc von konvexen Körpern, Math. Ann. 122 (1950), 205220.

[J1] F. John . Polar correspondence with respect to convex regions, Duke Math. J. 3 (1937), 355369.

[LM] J. Lindenstrauss and V. D. Milman , Local theory of normed spaces and convexity, Handbook of Convex Geometry ( P. M. Gruber and J. M. Wills , eds.), North-Holland, Amsterdam, 1993, pp. 11491220.

[MP] V. D. Milman and A. Pajor . Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. Geometric Aspects of Functional Analysis ( J. Lindenstrauss and V. D. Milman , eds.), vol. 1376, Springer Lecture Notes in Math., 1989, 64104.

[P1] C. M. Petty . Centroid surfaces. Pacific J. Math. 11 (1961), 15351547.

[P2] C. M. Petty , Ellipsoids, Convexity and its Applications ( P. M. Gruber and J. M. Wills , eds.). Birkhäuser, Basel, 1983, pp. 264276.

[RT] C. A. Rogers and S. J. Taylor , The analysis of additive set functions in Euclidean space, Acta Math. 101 (1959), 273302.

[Sc] F. Scheck , Mechanics, Springer-Verlag, Berlin Heidelberg, 1990.

[Sh] G. C. Shephard , Shadow systems of convex bodies, Israel J. Math. 2 (1964), 229–36.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 8 *
Loading metrics...

Abstract views

Total abstract views: 64 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 28th May 2017. This data will be updated every 24 hours.