Estermann measure

In plane geometry the Estermann measure is a number defined for any bounded convex set describing how close to being centrally symmetric it is.

It is the ratio of areas between the given set and its smallest centrally symmetric convex superset.

is the center of symmetry of the smallest centrally-symmetric set containing a given convex body

[1][2] The curve of constant width with the smallest possible Estermann measure is the Reuleaux triangle.

[4][1][2] Subsequent proofs were given by Friedrich Wilhelm Levi, by István Fáry, and by Isaak Yaglom and Vladimir Boltyansky.

A Reuleaux triangle and its reflection enclosed by their smallest centrally symmetric convex superset, a regular hexagon