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On the reverse Lp–busemann–petty centroid inequality

  • Stefano Campi (a1) and Paolo Gronchi (a2)
  • DOI: http://dx.doi.org/10.1112/S0025579300016004
  • Published online: 01 February 2010
Abstract
Abstract

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.

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[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.

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Mathematika
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  • EISSN: 2041-7942
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