Uniqueness results for minimal enclosing ellipsoids. (English) Zbl 1172.52300

Schröcker, Hans-Peter

Comput. Aided Geom. Des. 25, No. 9, 756-762 (2008).

Summary: We prove uniqueness of the minimal enclosing ellipsoid with respect to strictly eigenvalue convex size functions. Special examples include the classic case of minimal volume ellipsoids (Löwner ellipsoids), minimal surface area ellipsoids or, more generally, ellipsoids that are minimal with respect to quermass integrals.

MSC:

52A40	Inequalities and extremum problems involving convexity in convex geometry
52A38	Length, area, volume and convex sets (aspects of convex geometry)
52A27	Approximation by convex sets
52A39	Mixed volumes and related topics in convex geometry

Keywords:

minimal ellipsoid; Löwner ellipsoid; quermass integral; mean cross section measure; surface area; arc-length

References:

[1]	Behrend, F., Über einige affininvarianten konvexer bereiche, Math. ann., 113, 712-747, (1937) · JFM 62.0834.03
[2]	Behrend, F., Über die kleinste umbeschriebene und die größte einbeschriebene ellipse eines konvexen bereiches, Math. ann., 115, 397-411, (1938) · JFM 64.0731.05
[3]	Bonnesen, T.; Fenchel, W., Theorie der konvexen Körper, (1971), Chelsea Publishing Company Bronx, New York · Zbl 0906.52001
[4]	Boyd, S.; Vandenberghe, L., Convex optimization, (2004), Cambridge University Press · Zbl 1058.90049
[5]	Busemann, H., The foundations of Minkowski geometry, Comment. math. helv., 24, 156-186, (1950)
[6]	Carlson, B.C., Some inequalities for hypergeometric functions, Proc. amer. math. soc., 17, 32-39, (1966) · Zbl 0137.26803
[7]	Danzer, L.; Laugwitz, D.; Lenz, H., Über das Löwnersche ellipsoid und sein analogon unter den einem eikörper einbeschriebene ellipsoiden, Arch. math., 8, 214-219, (1957) · Zbl 0078.35803
[8]	Davis, C., All convex invariant functions of Hermitian matrices, Arch. math., 8, 276-278, (1957) · Zbl 0086.01702
[9]	Firey, W.J., Some applications of means of convex bodies, Pacific J. math., 14, 1, 53-60, (1964) · Zbl 0126.38405
[10]	Fisher, D.D., Minimum ellipsoids, Math. comput., 18, 88, 669-673, (1964) · Zbl 0122.14401
[11]	Gruber, P.M., Application of an idea of Voronoi to John type problems, Adv. in math., 218, 2, 309-351, (2008) · Zbl 1144.52003
[12]	John, F., Extremum problems with inequalities as subsidiary conditions, (), 187-204
[13]	Kumar, P.; Yıldırım, E.A., Minimum volume enclosing ellipsoids and core sets, J. optim. theory appl., 126, 1, 1-21, (2005) · Zbl 1093.90039
[14]	Lehmer, D.H., Approximations to the area of an n-dimensional ellipsoid, Canad. J. math., 2, 267-282, (1950) · Zbl 0037.17603
[15]	Lewis, A.S., Convex analysis on the Hermitian matrices, SIAM J. optim., 6, 1, 164-177, (1996) · Zbl 0849.15013
[16]	Muir, T., On the perimeter of an ellipse, Messenger math., 12, 149, (1883)
[17]	Nemirovski, A., Advances in convex optimization: conic programming, (), 413-444 · Zbl 1135.90379
[18]	Peano, G., Valori approssimati per l’area di un ellisoide, Rome, R. accad. dei lincei, rendiconti, 6, 2, 317-321, (1890) · JFM 22.0481.02
[19]	Pólya, G., Approximations to the area of the ellipsoid, Publicationes del instituto de matematica, rosario, 5, 1-13, (1943)
[20]	Santalo, L.A., Integral geometry and geometric probability, (2004), Cambridge University Press · Zbl 0063.06708
[21]	Schröcker, H.-P., Minimal enclosing hyperbolas of line sets, Beitr. algebra geom., 48, 2, 367-381, (2007) · Zbl 1167.53003
[22]	Tee, G., Surface area and capacity of ellipsoids in n dimensions, New Zealand J. math., 34, 2, 165-198, (2005) · Zbl 1115.41031
[23]	Todd, M.J.; Yıldırım, E.A., On Khachiyan’s algorithm for the computation of minimal volume enclosing ellipsoids, Discrete appl. math., 155, 13, 1731-1744, (2007) · Zbl 1151.90516