John ellipsoid

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In mathematics, the 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:

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 ci > 0 and unit vectors ui ∈ Sn−1 ∩ ∂K such that

\sum_{i = 1}^{m} c_{i} u_{i} = 0

and, for all x ∈ Rn

x = \sum_{i = 1}^{m} c_{i} (x \cdot u_{i}) u_{i}.

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