In mathematics, the John ellipsoid or Löwner-John ellipsoid E(K) associated to a convex body K in n-dimensional Euclidean space Rn is the ellipsoid of maximal n-dimensional volume contained within K. The John ellipsoid is named after the German mathematician Fritz John. The following refinement of John's original theorem, due to Ball (1992), gives necessary and sufficient conditions for the John ellipsoid of K to be the closed unit ball B of Rn:
and, for all x ∈ Rn
- Steiner inellipse, the special case of the John ellipsoid for a triangle
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