NDLI: John and Loewner Ellipsoids

Content Provider	Springer Nature Link
Author	Gruber, Peter M.
Copyright Year	2011
Abstract	John’s ellipsoid criterion characterizes the unique ellipsoid of globally maximum volume contained in a given convex body C. In this article local and global maximum properties of the volume on the space of all ellipsoids in C are studied, where ultra maximality is a stronger version of maximality: the volume is nowhere stationary. The ellipsoids for which the volume is locally maximum, resp. locally ultra maximum are characterized. The global maximum is the only local maximum and for generic C it is an ultra maximum. The characterizations make use of notions originating from the geometric theory of positive quadratic forms. Part of these results generalize to the case where the ellipsoids are replaced by affine copies of a convex body D. In contrast to the ellipsoid case, there are convex bodies C and D, such that on the space of all affine images of D in C the volume has countably many local maxima. All results have dual counterparts. Extensions to the surface area and, more generally, to intrinsic volumes are mentioned.
Starting Page	776
Ending Page	788
Page Count	13
File Format	PDF
ISSN	01795376
Journal	Discrete & Computational Geometry
Volume Number	46
Issue Number	4
e-ISSN	14320444
Language	English
Publisher	Springer-Verlag
Publisher Date	2011-05-11
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	John ellipsoid Loewner ellipsoid Eutaxy Perfection Maximum volume Convex body Computational Mathematics and Numerical Analysis Combinatorics
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science Computational Theory and Mathematics Geometry and Topology

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