John's theorem states that if 
is a compact set
that is convex and has nonempty interior,
and its John ellipsoid is the unit
ball 
,
then there are contact points 
and positive numbers
such that
 |
Conversely, the existence of such points and numbers characterizes 
as the unique maximum-volume ellipsoid contained in 
. Here 
is the 
identity matrix,
and at most 
contact points are needed (John 1948, Henk 2012).
The theorem also gives
 |
If 
is a centrally
symmetric set, the factor 
can be replaced by 
; both factors are best possible (John 1948, Henk 2012).
