John's Theorem
John's theorem states that if K subset R^n is a compact set that is convex and has nonempty interior, and its John ellipsoid is the unit ball B^n, then 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 unique maximum-volume ellipsoid contained in K. Here I_n is the n×n identity matrix, and at most n(n+3)/2 contact points are needed (John 1948, Henk 2012).
The theorem also gives
| B^n subset= K subset= nB^n. |
If K is a centrosymmetric set, the factor n can be replaced by sqrt(n); both factors are best possible (John 1948, Henk 2012).
See also
John Ellipsoid, Löwner Ellipsoid, Unit BallExplore with Wolfram|Alpha
More things to try:
References
Henk, M. "Löwner-John Ellipsoids." In Optimization Stories. Doc. Math. Extra Vol. ISMP, 95-106, 2012. https://doi.org/10.4171/DMS/6/15.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's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JohnsTheorem.html