Farthest points on curves and curves of constant width

In this talk we will investigate two simple properties of closed curves in the plane:

   (a) for every point x on the curve, there is a unique point on the curve which is farthest from x,

   (b) if y is the farthest point from x, then x is the unique farthest point from y.

We’ll use elementary differential geometry to describe the curves with property (a) in terms of the relationship between the curve and its evolute and to show that any curve with property (b) is a curve of constant width.