The John ellipsoid of a convex body 
is the unique ellipsoid
of maximum volume contained in 
. Here a convex body is a compact
set that is convex and has nonempty interior.
In two dimensions, the John ellipsoid is also called the John ellipse and has maximum
area among ellipses contained in 
(John 1948, Ball 1992, Henk 2012).
After an affine transformation, suppose that the John ellipsoid is the unit ball 
. John's theorem states
that there are contact points 
and positive numbers
such that
 |
Conversely, the existence of such points and numbers characterizes 
as the John ellipsoid. Here 
is the 
identity matrix,
and at most 
contact points are needed (John 1948, Henk 2012).
In this normalization, John's theorem also gives
 |
If 
is a centrally symmetric set, the factor
can be replaced by 
; both factors are best possible (John 1948, Henk 2012).
The Steiner inellipse is the John ellipse of a triangle. The unique maximal-area inellipse of a convex quadrilateral is similarly its John ellipse (Horwitz 2005).
