Constant width
Named by 5 essays across one field — each of them below, with the objects they name alongside it.
Round is not the only way to be the same width
A shape that measures the same in every direction sounds like a description of a circle. It is not — there are infinitely many others, one of them is on a coin in most people's pockets, and a drill built from one cuts a nearly square hole.
The most area a fence can hold
One length of boundary, and the question of what shape to bend it into. The answer is a circle, everybody knows it, and the argument that convinced the nineteenth century turned out to prove something slightly different.
The shape described from outside
A convex shape can be given by its boundary or by the family of lines that touch it, and the second description turns the constant-width condition into one line of arithmetic — after which the perimeter falls out, and curves with no corners at all can simply be written down.
The least area a width can hold
Barbier's theorem says every curve of constant width has the same perimeter, which removes perimeter as a way of telling the family apart. Area is not like that — the circle holds the most and the Reuleaux triangle the least — and the reason the minimiser has corners is a constraint rather than a preference.
The same question in space
Intersect four balls at the corners of a tetrahedron and the result is not of constant width — it misses by two and a half per cent, computed exactly. Repairing it gives a body that is, and whether that body is the smallest of its kind has been open for a century.
Named alongside it
The objects these essays reach for when they reach for this one.
ConvexityReuleaux triangleSupport functionAreaBarbier's theoremCircleConvex hullCurvatureDualityExistence proofExtremal problemFourier series