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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: saroglou@math.uoc.gr)
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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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