plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

the Kovner–Besicovitch measure is a number defined for any bounded

convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...

describing how close to being

centrally symmetric In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or ...

it is. It is the fraction of the area of the set that can be covered by its largest centrally symmetric subset.

Properties

This measure 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 transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...

s of the plane. If

c

is the center of symmetry of the largest centrally-symmetric set within a given convex body

K

, then the centrally-symmetric set itself is the intersection of

K

with its reflection across

c

Minimizers

The convex sets with the smallest possible Kovner–Besicovitch measure are the triangles, for which the measure is 2/3. The result that triangles are the minimizers of this measure is known as Kovner's theorem or the Kovner–Besicovitch theorem, and the inequality bounding the measure above 2/3 for all convex sets is the Kovner–Besicovitch inequality. The

curve of constant width In geometry, a curve of constant width is a simple closed curve in the plane (geometry), plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of ...

with the smallest possible Kovner–Besicovitch measure is the Reuleaux triangle.

Computational complexity

The Kovner–Besicovitch measure of any given convex polygon with

n

vertices can be found in time

O(n\log n)

by determining a translation of the reflection of the polygon that has the largest possible overlap with the unreflected polygon.

History

Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentcalculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

Mikhail Lavrentyev Mikhail Alekseyevich Lavrentyev (or Lavrentiev, ; November 19, 1900 – October 15, 1980) was a Soviet mathematician and hydrodynamicist. Early years Lavrentyev was born in Kazan, where his father was an instructor at a college (he later became ...

and Lazar Lyusternik, where it was credited to Soviet mathematician and geophysicist . Additional proofs were given by Abram Samoilovitch Besicovitch and by

István Fáry István Fáry (30 June 1922 – 2 November 1984) was a Hungarian-born mathematician known for his work in geometry and algebraic topology.. He proved Fáry's theorem that every planar graph has a straight-line embedding in 1948, and the Fáry� ...

, who also proved that every minimizer of the Kovner–Besicovitch measure is a triangle.

References

External links