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.

Hide All
[B]Blaschke W.. Affine Geometric XIV: Eine Minimumaufgabc für Legendres Trägheitsellipsoid, Ber. Verh. Sächs. Akad. Leipzig, Math.-Phys. Ki. 70 (1918), 7275.
[BB]Bisztriczky T. and Böröczky K. Jr., About the centroid body and the ellipsoid of inertia. Mathematika, to appear.
[CCG]Campi S.. Colesanti A. and Gronchi P., A note on Sylvester's problem for random polytopcs in a convex body. Rend. 1st. Mat. Univ. Trieste 31 (1999), 7994.
[CG]Campi S. and Gronchi P., The Lp-Busemann-Petty centroid inequality body, Adv. Math. 167 (2002), 128141.
[Fa]Fáry I.. Sur la dénsit des réscaux de domaines convexes, Bull. Soc. Math. France 78 (1950). 152161.
[FR]Fáry I. and Rédei L.. Der zentralsymmetrische Kern und die zentralsymmetrische Hüllc von konvexen Körpern, Math. Ann. 122 (1950), 205220.
[Fi]Firey W. J., p-means of convex bodies. Math. Scand. 10 (1962), 1724.
[G]Gardner R. J., Geometric Tomography, Cambridge University Press, Cambridge, 1995.
[J1]John F.. Polar correspondence with respect to convex regions, Duke Math. J. 3 (1937), 355369.
[J2]John F.. Extremum problems with inequalities as subsidiary conditions, In Courant Anniversary Volume (Interscience, New York), 1948, 187204.
[LM]Lindenstrauss J. and Milman V. D., Local theory of normed spaces and convexity, Handbook of Convex Geometry (Gruber P. M. and Wills J. M., eds.), North-Holland, Amsterdam, 1993, pp. 11491220.
[L]Lutwak E., The Brunn-Minkowski Firey theory I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131150.
[LYZ]Lutwak E., Yang D. and Zhang G., Lp affine isoperimetric inequalities, J. Differential Geom. 56 (2000), 111132.
[LZ]Lutwak E. and Zhang G., Blaschke Santaló inequalities, J. Differential Geom. 47 (1997), 116.
[MP]Milman V. D. and Pajor A.. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. Geometric Aspects of Functional Analysis (Lindenstrauss J. and Milman V. D., eds.), vol. 1376, Springer Lecture Notes in Math., 1989, 64104.
[P1]Petty C. M.. Centroid surfaces. Pacific J. Math. 11 (1961), 15351547.
[P2]Petty C. M., Ellipsoids, Convexity and its Applications (Gruber P. M. and Wills J. M., eds.). Birkhäuser, Basel, 1983, pp. 264276.
[RS]Rogers C. A. and Shephard G. C., Some extremal problems for convex bodies, Mathematika 5 (1958), 93102.
[RT]Rogers C. A. and Taylor S. J., The analysis of additive set functions in Euclidean space, Acta Math. 101 (1959), 273302.
[Sc]Scheck F., Mechanics, Springer-Verlag, Berlin Heidelberg, 1990.
[Sc]Schneider R., Convex bodies: the Brunn–Minkowski Theory, Cambridge University Press, Cambridge, 1993.
[Sh]Shephard G. C., 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: 9 *
Loading metrics...

Abstract views

Total abstract views: 77 *
Loading metrics...

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