Kakutani's theorem is a result in
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
named after
Shizuo Kakutani
was a Japanese and American mathematician, best known for his eponymous fixed-point theorem.
Biography
Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. At one point he spent two years at the Institu ...
. It states that every
convex body
In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non- empty interior. Some authors do not require a non-empty interior, merely that the set is non-empty.
A convex body K is called symmetric if it ...
in 3-
dimension
In physics and mathematics, the dimension of a mathematical 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 (1D) because only one coo ...
al space has a circumscribed
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
, i.e. a cube all of whose faces touch the body. The result was further generalized by
Yamabe and Yujobô to higher dimensions, and by Floyd to other circumscribed
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square.
Three equiva ...
s.
References
Theorems in convex geometry
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