John ellipsoid

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 c_i > 0 and unit vectors u_i ∈ Sⁿ⁻¹ ∩ ∂K such that

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

and, for all x ∈ Rⁿ

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

References

Ball, Keith M. (1992). "Ellipsoids of maximal volume in convex bodies". Geom. Dedicata 41 (2): 241–250. doi:10.1007/BF00182424. ISSN 0046-5755.
John, Fritz. "Extremum problems with inequalities as subsidiary conditions". Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, 187—204. Interscience Publishers, Inc., New York, N. Y., 1948. OCLC 1871554 MR 30135
Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.) 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. ISSN 0273-0979.