Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-28T13:41:27.771Z Has data issue: false hasContentIssue false
  • English
  • Français

VOLUMES OF PROJECTION BODIES OF SOME CLASSES OF CONVEX BODIES

Part of: General convexity

Published online by Cambridge University Press:  20 June 2011

Christos Saroglou
Christos Saroglou*
Affiliation:
Department of Mathematics, University of Crete, Greece (email: saroglou@math.uoc.gr)
Get access

Abstract

Schneider posed the problem of determining the maximal value of the affine invariant ∣ΠK∣/∣K∣d−1, where ΠK is the projection body of the d-dimensional convex body K. Some three-dimensional conjectures of Brannen, related to Schneider’s problem, are confirmed. Namely, we determine the maximal value of ∣ΠK∣/∣K∣2 in the class of three-dimensional zonoids, cones and double cones. Equality cases are, also, investigated. Moreover, results related to a conjecture of Petty, concerning the minimal value of the above quantity, are obtained. In particular, we provide a negative answer to a question of Martini and Mustafaev.

MSC classification

Secondary: 52A40: Inequalities and extremum problems 52A15: Convex sets in $3$ dimensions (including convex surfaces) 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 57 , Issue 2 , July 2011 , pp. 329 - 353
Copyright
Copyright © University College London 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Busemann, H., Volume in terms of concurrent sections. Pacific J. Math. 3 (1953), 1–12.CrossRefGoogle Scholar
[2]Brannen, N. S., Volumes of projection bodies. Mathematika 43 (1996), 255–264.CrossRefGoogle Scholar
[3]Brannen, N. S., Three-dimensional projection bodies. Adv. Geom. 5 (2005), 1–13.CrossRefGoogle Scholar
[4]Campi, S., Colesanti, A. and Gronchi, P., A note on Sylvester’s problem for random polytopes in a convex body. Rend. Istit. Mat. Univ. Trieste 31 (1999), 79–94.Google Scholar
[5]Campi, S. and Gronchi, P., Extremal convex sets for Sylvester–Busemann type functionals. Appl. Anal. 85 (2006), 129–141.CrossRefGoogle Scholar
[6]Gardner, R. J., Geometric Tomography (Encyclopedia of Mathematics and its Applications 58), Cambridge University Press (Cambridge, 1995).Google Scholar
[7]Lutwak, E., On a conjectured projection inequality of Petty. Contemp. Math. 113 (1990), 171–182.CrossRefGoogle Scholar
[8]Lutwak, E., On quermassintegrals of mixed projection bodies. Geom. Dedicata 33 (1990), 51–58.CrossRefGoogle Scholar
[9]Lutwak, E., Yang, D. and Zhang, G., A new affine invariant for polytopes and Schneider’s projection problem. Trans. Amer. Math. Soc. 353 (2001), 1767–1779.CrossRefGoogle Scholar
[10]Makai, E. Jr and Martini, H., The cross-section body, plane sections of convex bodies and approximation of convex bodies. I. Geom. Dedicata 63 (1996), 267–296.CrossRefGoogle Scholar
[11]Martini, H. and Mustafaev, Z., On isoperimetric inequalities in Minkowski spaces. J. Inequal. Appl. 2010 (2010), 18, Article ID 697954.CrossRefGoogle Scholar
[12]Milman, V. D. and Pajor, A., Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics 1376), Springer (Berlin, 1989), 64–104.CrossRefGoogle Scholar
[13]Petty, C. M., Centroid surfaces. Pacific J. Math. 11 (1961), 1535–1547.CrossRefGoogle Scholar
[14]Petty, C. M., Projection bodies. In Proc. Colloquium on Convexity (Copenhagen, 1965), Kobenhavns Univ. Mat. Inst. (Copenhagen, 1967), 234–241.Google Scholar
[15]Petty, C. M., Isoperimetric problems. In Proceedings of the Conference on Convexity and Combinatorial Geometry (Norman, OK, 1971), Dept. Math., Univ. Oklahoma (Norman, OK, 1971), 26–41.Google Scholar
[16]Reisner, S., Random polytopes and the volume product of symmetric convex bodies. Math. Scand. 57 (1985), 386–392.CrossRefGoogle Scholar
[17]Reisner, S., Zonoids with minimal volume product. Math. Z. 192 (1986), 339–346.CrossRefGoogle Scholar
[18]Rogers, C. A. and Shephard, G. C., The difference body of a convex body. Arch. Math. 8 (1957), 220–233.CrossRefGoogle Scholar
[19]Schneider, R., Random hyperplanes meeting a convex body. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 61 (1982), 379–387.CrossRefGoogle Scholar
[20]Schneider, R., Convex Bodies: The Brunn–Minkowski Theory (Encyclopedia of Mathematics and its Applications 44), Cambridge University Press (Cambridge, 1993).CrossRefGoogle Scholar
[21]Weil, W., Über die Projektionenkörper konvexer Polytope. Arch. Math. 22 (1971), 664–672.CrossRefGoogle Scholar
[22]Weil, W., Kontinuierliche Linearkombination won Strecken. Math. Z. 148 (1976), 71–84.CrossRefGoogle Scholar
[23]Zhang, G., Restricted chord projection and affine inequalities. Geom. Dedicata 39 (1991), 213–222.CrossRefGoogle Scholar