Convex body
Encyclopedia
In mathematics
, a convex body in n-dimension
al 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. Symmetric convex bodies are in a one-to-one correspondence
with the unit balls of norms
on Rn.
Important examples of convex bodies are the Euclidean ball, the hypercube
and the cross-polytope
.
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...
, a convex body 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 is a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
convex set
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
with non-empty
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
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....
.
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
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
its antipode
Antipode
Antipode, Antipodes, or Antipodeans may refer to:* Antipodal point, the diametrically opposite point on a sphere* Antipodes Water Company, a premium bottled water brand...
, −x, also lies in K. Symmetric convex bodies are in a one-to-one correspondence
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
with the unit balls of norms
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
on Rn.
Important examples of convex bodies are the Euclidean ball, the hypercube
Hypercube
In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
and the cross-polytope
Cross-polytope
In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of . The cross-polytope is the convex hull of its vertices...
.