John ellipsoid explained

In mathematics, the John ellipsoid or Löwner–John ellipsoid associated to a convex body in -dimensional Euclidean space can refer to the -dimensional ellipsoid of maximal volume contained within or the ellipsoid of minimal volume that contains .

Often, the minimal volume ellipsoid is called the Löwner ellipsoid, and the maximal volume ellipsoid is called the John ellipsoid (although John worked with the minimal volume ellipsoid in his original paper).^[1] One can also refer to the minimal volume circumscribed ellipsoid as the outer Löwner–John ellipsoid, and the maximum volume inscribed ellipsoid as the inner Löwner–John ellipsoid.^[2]

The German-American mathematician Fritz John proved in 1948 that each convex body in is circumscribed by a unique ellipsoid of minimal volume, and that the dilation of this ellipsoid by factor is contained inside the convex body.^[3] That is, the outer Lowner-John ellipsoid is larger than the inner one by a factor of at most . For a balanced body, this factor can be reduced to

Properties

The inner Löwner–John ellipsoid of a convex body

is a closed unit ball in if and only if and there exists an integer and, for, real numbers and unit vectors (where is the unit n-sphere) such that^[4]

$$\backslash sum\_^m\; c\_i\; u\_i\; =\; 0$$

and, for all

$$x\; =\; \backslash sum\_^m\; c\_i\; (x\; \backslash cdot\; u\_i)\; u\_i.$$

Computation

In general, computing the John ellipsoid of a given convex body is a hard problem. However, for some specific cases, explicit formulas are known. Some cases are particularly important for the ellipsoid method.

Let be an ellipsoid in defined by a matrix and center . Let be a nonzero vector in Let be the half-ellipsoid derived by cutting at its center using the hyperplane defined by . Then, the Lowner-John ellipsoid of is an ellipsoid defined by:where is a vector defined by:$$\backslash mathbf\; =\; \backslash frac\; \backslash mathbf$$Similarly, there are formulas for other sections of ellipsoids, not necessarily through its center.

Applications

The computation of Löwner–John ellipsoids (and in more general, the computation of minimal-volume polynomial level sets enclosing a set) has found many applications in control and robotics.^[5] In particular, computing Löwner–John ellipsoids has applications in obstacle collision detection for robotic systems, where the distance between a robot and its surrounding environment is estimated using a best ellipsoid fit.^[6]

Löwner–John ellipsoids has also been used to approximate the optimal policy in portfolio optimization problems with transaction costs.^[7]

See also

• • Ellipsoid method
• Steiner inellipse, the special case of the inner Löwner–John ellipsoid for a triangle.
• Fat object, related to radius of largest contained ball.

References

• Gardner . Richard J. . The Brunn-Minkowski inequality . Bull. Amer. Math. Soc. (N.S.) . 39 . 3 . 2002 . 355 - 405 (electronic) . 0273-0979 . 10.1090/S0273-0979-02-00941-2. free .

Notes and References

1. Güler. Osman. Gürtuna. Filiz. 2012. Symmetry of convex sets and its applications to the extremal ellipsoids of convex bodies. Optimization Methods and Software. en. 27. 4–5. 735–759. 10.1080/10556788.2011.626037. 2971340 . 1055-6788. 10.1.1.664.6067.
2. Book: Ben-Tal, A.. Lectures on modern convex optimization : analysis, algorithms, and engineering applications. 2001. Society for Industrial and Applied Mathematics. Nemirovskiĭ, Arkadiĭ Semenovich.. 0-89871-491-5. Philadelphia, PA. 46538510.
3. 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.
4. Ball. Keith M.. 1992. Ellipsoids of maximal volume in convex bodies. Geom. Dedicata. 41. 2. 241 - 250. math/9201217. 10.1007/BF00182424. 18330466 . 0046-5755.
5. Dabbene . Fabrizio . Henrion . Didier . Lagoa . Constantino M. . 2017 . Simple approximations of semialgebraic sets and their applications to control . Automatica . en . 78 . 110–118 . 1509.04200 . 10.1016/j.automatica.2016.11.021. 5778355 .
6. Rimon . Elon . Boyd . Stephen . 1997 . Obstacle Collision Detection Using Best Ellipsoid Fit . Journal of Intelligent and Robotic Systems . 18 . 2 . 105–126 . 10.1023/A:1007960531949 . 10505238.
7. Shen. Weiwei. Wang. Jun. Transaction Costs-Aware Portfolio Optimization via Fast Lowner-John Ellipsoid Approximation . Proceedings of the AAAI Conference on Artificial Intelligence . 2015. https://web.archive.org/web/20170116192444/https://pdfs.semanticscholar.org/7b31/2141616f092137c12397a47d11d94ddcea78.pdf. dead. 2017-01-16. 29 . 1854–1860. 10.1609/aaai.v29i1.9453 . 14746495 .