Busemann's theorem
Encyclopedia
In mathematics
, Busemann's theorem is a theorem
in Euclidean geometry
and geometric tomography
. It was first proved by Herbert Busemann
in 1949 and was motivated by his theory of area in Finsler spaces.
in n-dimension
al Euclidean space
Rn containing the origin
in its interior
. Let S be an (n − 2)-dimensional linear subspace
of Rn. For each unit vector θ in S⊥, the orthogonal complement of S, let Sθ denote the closed (n − 1)-dimensional half-space containing θ and with S as its boundary
. Define r(θ) to be the (n − 1)-dimensional volume of K ∩ Sθ. Let C be the curve {θr(θ)} in S⊥. Then C forms the boundary of a convex body in S⊥.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Busemann's theorem is a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
in Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
and geometric tomography
Geometric tomography
Geometric tomography is a mathematical field that focuses on problems of reconstructing homogeneous objects from tomographic data . More precisely, according to R.J...
. It was first proved by Herbert Busemann
Herbert Busemann
Herbert Busemann was a German-American mathematician specializing in convex and differential geometry. He is the author of Busemann's theorem in Euclidean geometry and geometric tomography. He was a member of the Royal Danish Academy and a winner of the Lobachevsky Medal , the first American...
in 1949 and was motivated by his theory of area in Finsler spaces.
Statement of the theorem
Let K be a convex bodyConvex 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 n-dimension
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...
al Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn containing the origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
in its interior
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
. Let S be an (n − 2)-dimensional linear subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
of Rn. For each unit vector θ in S⊥, the orthogonal complement of S, let Sθ denote the closed (n − 1)-dimensional half-space containing θ and with S as its boundary
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
. Define r(θ) to be the (n − 1)-dimensional volume of K ∩ Sθ. Let C be the curve {θr(θ)} in S⊥. Then C forms the boundary of a convex body in S⊥.