Editing Reuleaux triangle (section)

=== Yanmouti sets ===
The Yanmouti sets are defined as the [[convex hull]]s of an equilateral triangle together with three circular arcs, centered at the triangle vertices and spanning the same angle as the triangle, with equal radii that are at most equal to the side length of the triangle. Thus, when the radius is small enough, these sets degenerate to the equilateral triangle itself, but when the radius is as large as possible they equal the corresponding Reuleaux triangle. Every shape with width ''w'', diameter ''d'', and inradius ''r'' (the radius of the largest possible circle contained in the shape) obeys the inequality
:<math>w - r \le \frac{d}{\sqrt 3},</math>
and this inequality becomes an equality for the Yanmouti sets, showing that it cannot be improved.<ref>{{citation
 | last = Hernández Cifre | first = M. A.
 | doi = 10.2307/2695582
 | issue = 10
 | journal = [[American Mathematical Monthly]]
 | mr = 1806918
 | pages = 893–900
 | title = Is there a planar convex set with given width, diameter, and inradius?
 | volume = 107
 | year = 2000| jstor = 2695582
 }}.</ref>