John ellipsoid

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

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 its 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]

Properties

The John ellipsoid is named after the German-American mathematician Fritz John, who proved in 1948 that each convex body in Rn is circumscribed by a unique ellipsoid of minimal volume and that the dilation of this ellipsoid by factor 1/n is contained inside the convex body.[3]

The inner Löwner-John ellipsoid E(K) of a convex body K ⊂ Rn is a closed unit ball B in Rn 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[4]

and, for all x ∈ Rn

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
  • Banach–Mazur compactum – Set of n-dimensional subspaces of a normed space made into a compact metric space.
  • Steiner inellipse, the special case of the inner Löwner-John ellipsoid for a triangle.
  • Fat object, related to radius of largest contained ball.

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. 27 (4–5): 735–759. doi:10.1080/10556788.2011.626037. ISSN 1055-6788. S2CID 2971340.
  2. Ben-Tal, A. (2001). Lectures on modern convex optimization : analysis, algorithms, and engineering applications. Nemirovskiĭ, Arkadiĭ Semenovich. Philadelphia, PA: Society for Industrial and Applied Mathematics. ISBN 0-89871-491-5. OCLC 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. OCLC 1871554 MR30135
  4. Ball, Keith M. (1992). "Ellipsoids of maximal volume in convex bodies". Geom. Dedicata. 41 (2): 241–250. arXiv:math/9201217. doi:10.1007/BF00182424. ISSN 0046-5755. S2CID 18330466.
  5. Dabbene, Fabrizio; Henrion, Didier; Lagoa, Constantino M. (2017). "Simple approximations of semialgebraic sets and their applications to control". Automatica. 78: 110–118. arXiv:1509.04200. doi:10.1016/j.automatica.2016.11.021.
  6. Rimon, Elon; Boyd, Stephen (1997). "Obstacle Collision Detection Using Best Ellipsoid Fit". Journal of Intelligent and Robotic Systems. 18 (2): 105–126. doi:10.1023/A:1007960531949. S2CID 10505238.
  7. Shen, Weiwei; Wang, Jun (2015). "Transaction Costs-Aware Portfolio Optimization via Fast Lowner-John Ellipsoid Approximation" (PDF). Proceedings of the AAAI Conference on Artificial Intelligence. 29: 1854–1860. doi:10.1609/aaai.v29i1.9453. S2CID 14746495. Archived from the original (PDF) on 2017-01-16.
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