Skip to main content Accessibility help


  • René Brandenberg (a1) and Stefan König (a2)

Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the convex body. Since these coefficients are bounded by the dimension but possibly smaller, our inequalities sharpen the original ones. Since they can often be computed efficiently, the improved bounds may also be used to obtain better bounds in approximation algorithms.

Hide All
1.Alexander, R., The width and diameter of a simplex. Geom. Dedicata 6(1) 1977, 8794.
2.Ball, K., Ellipsoids of maximal volume in convex bodies. Geom. Dedicata 41(2) 1992, 241250.
3.Ball, K., An elementary introduction to modern convex geometry. In Flavors of Geometry, Cambridge University Press (Cambridge, 1997), 158.
4.Belloni, A. and Freund, R. M., On the symmetry function of a convex set. Math. Program. 111(1–2) 2008, 5793.
5.Betke, U. and Henk, M., Estimating sizes of a convex body by successive diameters and widths. Mathematika 39(2) 1992, 247257.
6.Bohnenblust, H. F., Convex regions and projections in Minkowski spaces. Ann. of Math. (2) 39(2) 1938, 301308.
7.Boltyanski, V. and Martini, H., Jung’s theorem for a pair of Minkowski spaces. Adv. Geom. 6(4) 2006, 645650.
8.Bonnesen, T. and Fenchel, W., Theorie der Konvexen Körper, Springer (Berlin, 1974) . Engl. transl. in Theory of Convex Bodies, BCS Associates (Moscow, ID, 1987).
9.Böröczky, K. Jr, Hernández Cifre, M. and Salinas, G., Optimizing area and perimeter of convex sets for fixed circumradius and inradius. Monatsh. Math. 138(2) 2003, 95110.
10.Bottema, O., Djordjevic, R. Z., Janic, R. R., Mitrinović, D. S. and Vasić, P. M., Geometric Inequalities, Wolters-Noordhoff (Groningen, The Netherlands, 1969).
11.Brandenberg, R., Dattasharma, A., Gritzmann, P. and Larman, D., Isoradial bodies. Discrete Comput. Geom. 32(4) 2004, 447457.
12.Brandenberg, R. and König, S., No dimension-independent core-sets for containment under homothetics. Discrete Comput. Geom. 49(1) 2013, 321 (Special Issue on SoCG ’11).
13.Brandenberg, R. and Roth, L., New algorithms for k-center and extensions. J. Comb. Optim. 18(4) 2009, 376392.
14.Brandenberg, R. and Roth, L., Minimal containment under homothetics: a simple cutting plane approach. Comput. Appl. 48(2) 2011, 325340.
15.Brieden, A., Geometric optimization problems likely not contained in APX. Discrete Comput. Geom. 28(2) 2002, 201209.
16.Danzer, L., Grünbaum, B. and Klee, V., Helly’s Theorem and its relatives. In Convexity (Proceedings of Symposia in Pure Mathematics 7) (ed. Klee, V.), American Mathematical Society (1963), 101180.
17.Eaves, B. C. and Freund, R. M., Optimal scaling of balls and polyhedra. Math. Program. 23(1) 1982, 138147.
18.Eggleston, H. G., Convexity, Vol. 47. Cambridge University Press (Cambridge, New York, 1958).
19.Fejes Tóth, L., Lagerungen in der Ebene auf der Kugel und im Raum, Springer (Berlin, Heidelberg, 1953).
20.Freund, R. M. and Orlin, J. B., On the complexity of four polyhedral set containment problems. Math. Program. 33(2) 1985, 139145.
21.Gonzáles Merino, B., On the ratio between successive radii of a symmetric convex body. Math. Inequal. Appl. 16(2) 2013, 569576.
22.Gritzmann, P. and Klee, V., Inner and outer j-radii of convex bodies in finite-dimensional normed spaces. Discrete Comput. Geom. 7(1) 1992, 255280.
23.Gritzmann, P. and Klee, V., Computational complexity of inner and outer j-radii of polytopes in finite-dimensional normed spaces. Math. Program. 59(1) 1993, 163213.
24.Gritzmann, P. and Lassak, M., Estimates for the minimal width of polytopes inscribed in convex bodies. Discrete Comput. Geom. 4(1) 1989, 627635.
25.Gruber, P. and Schuster, F., An arithmetic proof of John’s ellipsoid theorem. Arch. Math. 85(1) 2005, 8288.
26.Grünbaum, B., Measures of symmetry for convex sets. In Convexity: Proceedings of Symposia in Pure Mathematics, Vol. 7 (ed. Klee, V.), American Mathematical Society (Providence, RI, 1963), 271284.
27.Guo, Q. and Kaijser, S., On the distance between convex bodies. Northeastern Math. J. 5(3) 1999, 323331.
28.Hadwiger, H., Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer (Berlin, Heidelberg, 1957).
29.Henk, M., A generalization of Jung’s Theorem. Geom. Dedicata 42(2) 1992, 235240.
30.Henk, M., Löwner–John ellipsoids. In Optimization Stories (Documenta Mathematica, Extra Volume ISMP (2012)) (ed. Grötschel, M.), Deutsche Mathematiker-Vereinigung (Berlin, 2012), 95106.
31.Henk, M. and Hernández Cifre, M., Intrinsic volumes and successive radii. J. Math. Anal. Appl. 343(2) 2008, 733742.
32.Hernández Cifre, M., Salinas, G., Pastor, J. A. and Segura, S., Complete systems of inequalities for centrally symmetric convex sets in the n-dimensional space. Arch. Inequal. Appl. 1 2003, 155167.
33.John, F., Extremum problems with inequalities as subsidiary conditions. In Studies and Essays: Presented to R. Courant on His 60th Birthday, January 8, 1948, Interscience (New York, 1948), 187204.
34.Jung, H. W. E., Über die kleinste Kugel, die eine räumliche Figur einschließt. J. Reine Angew. Math. 123 1901, 241257.
35.Kaijser, S. and Guo, Q., Approximations of convex bodies by convex bodies. Northeastern Math. J. 19(4) 2003, 323332.
36.Khachiyan, L. G. and Todd, M. J., On the complexity of approximating the maximal inscribed ellipsoid for a polytope. Math. Program. 61(1) 1993, 137159.
37.Knauer, C., König, S. and Werner, D., Fixed parameter complexity of norm maximization. Preprint, 2013, arXiv:1307.6414.
38.Leichtweiss, K., Zwei Extremalprobleme der Minkowski-Geometrie. Math. Z. 62(1) 1955, 3749.
39.Perel’man, G. Y., k-radii of a convex body. Sib. Math. J. 28(4) 1987, 665666.
40.Schneider, R., Convex bodies: The Brunn–Minkowski Theory, Cambridge University Press (Cambridge, New York, 1993).
41.Schneider, R., Stability for some extremal properties of the simplex. J. Geom. 96(1) 2009, 135148.
42.Scott, P. R. and Awyong, P. W., Inequalities for convex sets. J. Inequal. Pure Appl. Math. 1(1) 2000, 113.
43.Steinhagen, P., Über die größte Kugel in einer konvexen Punktmenge. Abh. Math. Semin. Univ. 1(1) 1922, 1526.
44.Todd, M. J. and Yıldırım, E. A., On Khachiyan’s algorithm for the computation of minimum-volume enclosing ellipsoids. Discrete Appl. Math. 155(13) 2007, 17311744.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed