Kakutani's theorem is a result in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
named after
Shizuo Kakutani
was a Japanese-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 Institute for ...
. 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.
A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...
in 3-
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al space has a circumscribed
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. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
, 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. In Euclidea ...
s.
References
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Theorems in convex geometry
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