The n-diameter of planar sets of constant width

Ibragimov, Zair; Le, Tuan

doi:10.2140/involve.2012.5.327

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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

DOI: 10.2140/involve.2012.5.327

Abstract

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

d_{n} (E) = max (\prod_{1 \leq i < j \leq n} | z_{i} - z_{j} |)^{\frac{2}{n (n - 1)}},

where the maximum is taken over all $z_{k} \in E$ , $k = 1, 2, \dots, n$ . 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.

Keywords

$n$-diameter, constant width sets, Pólya extremal problem

Mathematical Subject Classification 2010

Primary: 30C65

Secondary: 05C25

Milestones

Received: 12 September 2011

Revised: 14 December 2011

Accepted: 16 December 2011

Published: 14 April 2013

Communicated by Michael Dorff

Authors

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

Vol. 5, No. 3, 2012