The Löwner ellipsoid of a convex body 
is the unique ellipsoid
of minimum volume containing 
. It is therefore also called the minimum-volume enclosing
ellipsoid. Here a convex body is a compact set that
is convex and has nonempty interior.
In two dimensions, the Löwner ellipsoid is called the Löwner ellipse. The
terminology honors Karel Löwner, who discovered the uniqueness of the minimum-volume
containing ellipsoid but did not publish the result (Henk 2012, O'Connor and Robertson).
After an affine transformation, suppose that the Löwner ellipsoid is the unit ball 
. A form of 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 Löwner ellipsoid. Here 
is the 
identity matrix,
and at most 
contact points are needed (John 1948, Henk 2012).
In this normalization,
 |
If 
is a centrally symmetric set, the factor
can be replaced by 
;
both factors are best possible (John 1948, Henk 2012).
The Löwner ellipse of a convex quadrilateral need not pass through all four vertices, so it need not be a circumellipse. If it does pass through all four vertices, it is the unique minimum-area circumellipse of the quadrilateral (Horwitz 2010).
