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. It is one for a set that is centrally symmetric, and less than one for sets whose closure is not centrally symmetric. It is invariant under affine transformations of the plane. [1]
If is the center of symmetry of the smallest centrally-symmetric set containing a given convex body , then the centrally-symmetric set itself is the convex hull of the union of with its reflection across . [1]
The shapes of minimum Estermann measure are the triangles, for which this measure is 1/2. [1] [2] The curve of constant width with the smallest possible Estermann measure is the Reuleaux triangle. [3]
The Estermann measure is named after Theodor Estermann, who first proved in 1928 that this measure is always at least 1/2, and that a convex set with Estermann measure 1/2 must be a triangle. [4] [1] [2] Subsequent proofs were given by Friedrich Wilhelm Levi, by István Fáry, and by Isaak Yaglom and Vladimir Boltyansky. [1]
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Theodor Estermann was a mathematician, working in the field of analytic number theory. The Estermann measure, a measure of the central symmetry of a convex set in the Euclidean plane, is named after him.
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In the geometry of the Euclidean plane, axiality is a measure of how much axial symmetry a shape has. It is defined as the ratio of areas of the largest axially symmetric subset of the shape to the whole shape. Equivalently it is the largest fraction of the area of the shape that can be covered by a mirror reflection of the shape.
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In geometry, a Reinhardt polygon is an equilateral polygon inscribed in a Reuleaux polygon. As in the regular polygons, each vertex of a Reinhardt polygon participates in at least one defining pair of the diameter of the polygon. Reinhardt polygons with sides exist, often with multiple forms, whenever is not a power of two. Among all polygons with sides, the Reinhardt polygons have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter. They are named after Karl Reinhardt, who studied them in 1922.
