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 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 a closed unit ball B in 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 ci > 0 and unit vectors ui ∈ Sn−1 ∩ ∂K such that
and, for all x ∈ Rn
Applications
- Obstacle Collision Detection [1]
See also
- Steiner inellipse, the special case of the John ellipsoid for a triangle
References
- ↑ Rimon, Elon; Boyd, Stephen. "Obstacle Collision Detection Using Best Ellipsoid Fit". Journal of Intelligent and Robotic Systems. 18: 105–126.
- 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.
This article is issued from Wikipedia - version of the 11/30/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.