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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: [email protected])
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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

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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