The Lowner–John ellipsoid is a minimum volume enclosing ellipsoid of some convex body $K$. This ellipsoid is unique (as is the maximum volume ellipsoid contained in $K$). I'm looking for algorithms based on those ellipsoids, or any significant practical use that requires finding them explicitly.
Rimon and Boyd use Lowner–John ellipsoids in their "...method for estimating the distance between a robot and its surrounding environment".
From the abstract:
First, approximate the detailed geometry of the robot and its environment by minimum-volume enclosing ellipsoids. ... Then compute a conservative distance estimate using ... [a] formula for the distance of a point from an n-dimensional ellipse.