John ellipsoid

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.

In 1948, Fritz John proved that each convex body in Rⁿ contains a unique ellipsoid of maximal volume. Thus, each convex body has an affine image whose ellipsoid of maximal volume is the Euclidean unit ball. He also gave necessary and sufficient conditions for this ellipsoid to be a ball.

The following refinement of John's original theorem, due to Keith Ball, gives necessary and sufficient conditions for the John ellipsoid of K to be a closed unit ball B in Rⁿ:

The John ellipsoid E(K) of a convex body K ⊂ Rⁿ is B if and only if B ⊆ K and there exists an integer m ≥ n and, for i = 1, ..., m, real numbers c_i > 0 and unit vectors u_i ∈ Sⁿ⁻¹ ∩ ∂K such that

and, for all x ∈ Rⁿ

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