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Bulletin of the London Mathematical Society Advance Access originally published online on May 9, 2008
Bulletin of the London Mathematical Society 2008 40(4):604-612; doi:10.1112/blms/bdn039
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© 2008 London Mathematical Society

A Favard-type problem for 3d convex bodies

Stefano Campi

Dipartimento di Ingegneria dell’Informazione
Università degli Studi di Siena
Via Roma 56
53100 Siena
Italy

Paolo Gronchi

Dipartimento di Matematica e Applicazioni per l’Architettura
Università degli Studi di Firenze
Piazza Ghiberti 27
50122 Firenze
Italy
paolo@fi.iac.cnr.it

Received 3 May 2006. Revision received 27 July 2007.

A theorem due to Favard states that among all plane sets of given area and perimeter, the symmetric lens has maximum circumradius. This paper deals with the higher-dimensional problem of finding the convex body in R3 of given volume and mean width with the largest possible diameter. It is shown that the solution is the convex hull of a surface of revolution with constant Gauss curvature and a segment lying on the axis of revolution. Such a body is conjectured also to maximize the circumradius in the same class.

2000 Mathematics Subject Classification 52A40 (primary), 52A38, 49Q20 (secondary).


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