Kakutani's theorem (geometry)
Encyclopedia
Kakutani's theorem is a result in geometry
named after Shizuo Kakutani
. It states that every convex body
in 3-dimensional
space has a circumscribed cube
, i.e. a cube all whose faces touch the body. The result was further generalized by Yamabe
and Yujobô to higher dimension, and by Floyd to other circumscribed parallelepiped
s.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
named after Shizuo Kakutani
Shizuo Kakutani
was a Japanese-born American mathematician, best known for his eponymous fixed-point theorem.Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. Early in his career he spent two years at the Institute for Advanced Study in Princeton at the invitation of the...
. It states that every convex body
Convex body
In mathematics, a convex body in n-dimensional Euclidean space Rn is a compact convex set with non-empty interior.A convex body K is called symmetric if it is centrally symmetric with respect to the origin, i.e. a point x lies in K if and only if its antipode, −x, also lies in K...
in 3-dimensional
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
space has a circumscribed cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
, i.e. a cube all whose faces touch the body. The result was further generalized by Yamabe
Hidehiko Yamabe
-External links:...
and Yujobô to higher dimension, and by Floyd to other circumscribed parallelepiped
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...
s.