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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.

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    $\begingroup$ This seems like a rather broad reference request, maybe too broad for this platform. Can you narrow it down, say algorithms in a specific field? (You give some tags that don't seem to relate to the question body at all.) What does Google Scholar give you, and why is that not what you want? $\endgroup$ – Raphael Jun 19 '15 at 17:01
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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

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