John Ellipsoid

The John ellipsoid of a convex body K subset R^n is the unique ellipsoid of maximum volume contained in K. Here a convex body is a compact set that is convex and has nonempty interior. In two dimensions, the John ellipsoid is also called the John ellipse and has maximum area among ellipses contained in K (John 1948, Ball 1992, Henk 2012).

After an affine transformation, suppose that the John ellipsoid is the unit ball B^n. John's theorem states that there are contact points u_i in partialK intersection partialB^n and positive numbers c_i such that

Conversely, the existence of such points and numbers characterizes B^n as the John ellipsoid. Here I_n is the n×n identity matrix, and at most n(n+3)/2 contact points are needed (John 1948, Henk 2012).

In this normalization, John's theorem also gives

If K is a centrosymmetric set, the factor n can be replaced by sqrt(n); both factors are best possible (John 1948, Henk 2012).

The Steiner inellipse is the John ellipse of a triangle. The unique maximal-area inellipse of a convex quadrilateral is similarly its John ellipse (Horwitz 2005).

See also

Ellipsoid, Inellipse, John's Theorem, Löwner Ellipsoid, Steiner Inellipse

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References

Ball, K. "Ellipsoids of Maximal Volume in Convex Bodies." Geom. Dedicata 41, 241-250, 1992. https://doi.org/10.1007/BF00182424.Henk, M. "Löwner-John Ellipsoids." In Optimization Stories. Doc. Math. Extra Vol. ISMP, 95-106, 2012. https://doi.org/10.4171/DMS/6/15.Horwitz, A. "Ellipses of Maximal Area and of Minimal Eccentricity Inscribed in a Convex Quadrilateral." Aust. J. Math. Anal. Appl. 2, 1-12, 2005.John, F. "Extremum Problems with Inequalities as Subsidiary Conditions." In Studies and Essays Presented to R. Courant on His 60th Birthday. New York: Interscience, pp. 187-204, 1948.

Cite this as:

Weisstein, Eric W. "John Ellipsoid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JohnEllipsoid.html

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