Algorithms that are using John's ellipsoid
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.


E. Rimon and S Boyd. Obstacle Collision Detection Using Best Ellipsoid Fit. Journal of Intelligent and Robotic Systems, 1997 - Springer