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.
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References
Ball, K.M.: Ellipsoids of maximal volume in convex bodies. Geom. Dedic. 41, 241–250 (1992)
Ball, K.: Convex geometry and functional analysis. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces I, pp. 161–194. North-Holland, Amsterdam (2001)
Bastero, J., Romance, M.: John’s decomposition of the identity in the non-convex case. Positivity 6, 1–16 (2002)
Coulangeon, R.: Spherical designs and zeta functions of lattices. Int. Math. Res. Not. Art. ID 49620, 16 (2006)
Danzer, L., Laugwitz, D., Lenz, H.: Über das Löwnersche Ellipsoid und sein Analogon unter den einem Eikörper einbeschriebenen Ellipsoiden. Arch. Math. 8, 214–219 (1957)
Giannopoulos, A.A., Milman, V.D.: Euclidean structure in finite dimensional normed spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces I, pp. 707–779. North-Holland, Amsterdam (2001)
Giannopoulos, A.A., Perissinaki, I., Tsolomitis, A.: John’s theorem for an arbitrary pair of convex bodies. Geom. Dedic. 84, 63–79 (2001)
Gordon, Y., Litvak, A.E., Meyer, M., Pajor, A.: John’s decomposition of the identity in the general case and applications. J. Differ. Geom. 68, 99–119 (2004)
Gruber, P.M.: Minimal ellipsoids and their duals. Rend. Circ. Mat. Palermo Suppl. 37(2), 35–64 (1988)
Gruber, P.M.: Convex and discrete geometry. In: Grundlehren Math. Wiss., vol. 336. Springer, Berlin (2007)
Gruber, P.M.: Application of an idea of Voronoĭ to John type problems. Adv. Math. 218, 309–351 (2008)
Gruber, P.M.: Application of an idea of Voronoi to lattice packing, in preparation
Gruber, P.M.: Application of an idea of Voronoi to lattice zeta functions, in preparation
Gruber, P.M.: Voronoi type criteria for lattice coverings with balls, in preparation
Gruber, P.M., Schuster, F.E.: An arithmetic proof of John’s ellipsoid theorem. Arch. Math. 85, 82–88 (2005)
Ji, L.: Exact fundamental domains for mapping class groups and equivariant cell decomposition for Teichmüller spaces via Minkowski reduction, manuscript 2009
John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8 (1948), pp. 187–204. Interscience, New York (1948)
Johnson, W.B., Lindenstrauss, J.: Basic concepts in the geometry of Banach spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces I, pp. 1–84. North-Holland, Amsterdam (2001)
Pełczyński, A.: Remarks on John’s theorem on the ellipsoid of maximal volume inscribed into a convex symmetric body in R n. Note Mat. 10(suppl. 2), 395–410 (1990)
Sarnak, P., Ströömbergsson, A.: Minima of Epstein’s zeta function and heights of flat tori. Invent. Math. 165, 115–151 (2006)
Schmutz, P.: Riemann surfaces with shortest geodesic of maximal length. Geom. Funct. Anal. 3, 564–631 (1993)
Schmutz Schaller (Schmutz), P.: Systoles on Riemann surfaces. Manuscr. Math. 85, 428–447 (1994)
Schmutz Schaller, P.: Geometry of Riemann surfaces based on closed geodesics. Bull. Am. Math. Soc. 35, 193–214 (1998)
Schmutz Schaller, P.: Perfect non-extremal Riemann surfaces. Can. Math. Bull. 43, 115–125 (2000)
Schneider, R.: Convex bodies: the Brunn–Minkowski theory. Cambridge University Press, Cambridge (1993)
Schröcker, H.-P.: Uniqueness results for minimal enclosing ellipsoids. Comput. Aided Geom. Des. 25, 756–762 (2008)
Voronoĭ (Voronoï; Woronoi), G.F.: Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Première mémoire: Sur quelques propriétés des formes quadratiques positives parfaites. J. Reine Angew. Math. 133, 97–178 (1908). Coll. Works II 171–238
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Gruber, P.M. John and Loewner Ellipsoids. Discrete Comput Geom 46, 776–788 (2011). https://doi.org/10.1007/s00454-011-9354-8
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DOI: https://doi.org/10.1007/s00454-011-9354-8