## Involve: A Journal of Mathematics

• Involve
• Volume 5, Number 3 (2012), 327-338.

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

#### 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

$d n ( E ) = max ∏ 1 ≤ i < j ≤ n | z i − z j | 2 n ( n − 1 ) ,$

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.

#### Article information

Source
Involve, Volume 5, Number 3 (2012), 327-338.

Dates
Revised: 14 December 2011
Accepted: 16 December 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513733501

Digital Object Identifier
doi:10.2140/involve.2012.5.327

Mathematical Reviews number (MathSciNet)
MR3044618

Zentralblatt MATH identifier
1277.30014

#### Citation

Ibragimov, Zair; Le, Tuan. The $n$-diameter of planar sets of constant width. Involve 5 (2012), no. 3, 327--338. doi:10.2140/involve.2012.5.327. https://projecteuclid.org/euclid.involve/1513733501

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