The width of a convex body S in n-dimension is the distance w between
2 parallel planes tangent to S. A convex body is said to be of constant width if w, which is a function on the (n-1)-sphere,
is constant. Most people have held
a 2-dimensional body of constant width in their hand at some point, since aspherical coins (e.g. the English 50 pence piece) must
be of constant width for vending machines to operate.
The problem of minimising the volume of a body of fixed constant width is referred to as the Blaschke-Lebesgue problem. While the plane
minimiser is long known to be the Reuleaux triangle, remarkably, the Blaschke-Lebesgue problem remains open in all dimension greater than 2.
Over the past few years myself and collaborators have considered constant width surfaces in 3-dimensions, identifying general properties
of the volume function , and considering a variational approach to the problem in the
rotationally symmetric case  and the full three dimensional case
Constant width curves arise in the potential theory of the furthest point distance function .