Vol. 5, No. 3, 2012

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The $n$-diameter of planar sets of constant width

Zair Ibragimov and Tuan Le

Vol. 5 (2012), No. 3, 327–338

We study the notion of n-diameter for sets of constant width. A convex set in the plane is said to be of constant width if the distance between two parallel support lines is constant, independent of the direction. The Reuleaux triangles are the well-known examples of sets of constant width that are not disks. The n-diameter of a compact set E in the plane is

dn(E) = max 1i<jn|zi zj| 2 n(n1) ,

where the maximum is taken over all zk E, k = 1,2,,n. We prove that if n = 5, then the Reuleaux n-gons have the largest n-diameter among all sets of given constant width. The proof is based on the solution of an extremal problem for n-diameter.

$n$-diameter, constant width sets, Pólya extremal problem
Mathematical Subject Classification 2010
Primary: 30C65
Secondary: 05C25
Received: 12 September 2011
Revised: 14 December 2011
Accepted: 16 December 2011
Published: 14 April 2013

Communicated by Michael Dorff
Zair Ibragimov
Department of Mathematics
California State University, Fullerton
McCarthy Hall 154
Fullerton, CA 92831
United States
Tuan Le
Department of Mathematical Sciences
Worcester Polytechnic Institute
100 Institute Road
Worcester, MA 01609
United States