John ellipsoid
Encyclopedia
In mathematics
, the John ellipsoid E(K) associated to a convex body
K in n-dimension
al Euclidean space
Rn is the ellipsoid of maximal n-dimensional volume
contained within K. The John ellipsoid is named after the German
mathematician
Fritz John
. The following refinement of John's original theorem, due to Ball (1992), gives necessary and sufficient conditions for the John ellipsoid of K to be the closed unit ball B of Rn:
The John ellipsoid E(K) of a convex body K ⊂ Rn is B if and only if
B ⊆ K and there exists an integer
m ≥ n and, for i = 1, ..., m, real number
s ci > 0 and unit vectors ui ∈ Sn−1 ∩ ∂K such that
and, for all x ∈ Rn
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...
, the John ellipsoid E(K) associated to a 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...
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 is the ellipsoid of maximal n-dimensional volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
contained within K. The John ellipsoid is named after the German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Fritz John
Fritz John
Fritz John was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation.-Biography:...
. The following refinement of John's original theorem, due to Ball (1992), gives necessary and sufficient conditions for the John ellipsoid of K to be the closed unit ball B of Rn:
The John ellipsoid E(K) of a convex body K ⊂ Rn is B 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....
B ⊆ K and there exists an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
m ≥ n and, for i = 1, ..., m, real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s ci > 0 and unit vectors ui ∈ Sn−1 ∩ ∂K such that
and, for all x ∈ Rn