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SUCCESSIVE RADII OF FAMILIES OF CONVEX BODIES

  • BERNARDO GONZÁLEZ (a1), MARÍA A. HERNÁNDEZ CIFRE (a2) and AICKE HINRICHS (a3)
Abstract

We study properties of the so-called inner and outer successive radii of special families of convex bodies. First we consider the balls of the $p$ -norms, for which we show that the precise value of the outer (inner) radii when $p\geq 2$ ( $1\leq p\leq 2$ ), as well as bounds in the contrary case $1\leq p\leq 2$ ( $p\geq 2$ ), can be obtained as consequences of known results on Gelfand and Kolmogorov numbers of identity operators between finite-dimensional normed spaces. We also prove properties that successive radii satisfy when we restrict to the families of the constant width sets and the $p$ -tangential bodies.

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[1]Ball, K., ‘Ellipsoids of maximal volume in convex bodies’, Geom. Dedicata 41 (1992), 241250.
[2]Betke, U. and Henk, M., ‘Estimating sizes of a convex body by successive diameters and widths’, Mathematika 39(2) (1992), 247257.
[3]Betke, U. and Henk, M., ‘A generalization of Steinhagen’s theorem’, Abh. Math. Semin. Univ. Hambg. 63 (1993), 165176.
[4]Betke, U., Henk, M. and Tsintsifa, L., ‘Inradii of simplices’, Discrete Comput. Geom. 17(4) (1997), 365375.
[5]Bonnesen, T. and Fenchel, W., Theory of Convex Bodies (eds. Boron, L., Christenson, C. and Smith, B.) (BCS Associates, Moscow, ID, 1987).
[6]Brandenberg, R., ‘Radii of convex bodies’, PhD Thesis, Technische Universität München, 2002.
[7]Brandenberg, R., ‘Radii of regular polytopes’, Discrete Comput. Geom. 33(1) (2005), 4355.
[8]Brandenberg, R., Dattasharma, A., Gritzmann, P. and Larman, D., ‘Isoradial bodies’, Discrete Comput. Geom. 32(4) (2004), 447457.
[9]Brandenberg, R. and Theobald, T., ‘Radii of simplices and some applications to geometric inequalities’, Beitr. Algebra Geom. 45(2) (2004), 581594.
[10]Carl, B. and Stephani, I., Entropy, Compactness and Approximation of Operators (Cambridge University Press, Cambridge, 1990).
[11]Chakerian, G. D. and Groemer, H., ‘Convex bodies of constant width’, in: Convexity and Its Applications (eds. Gruber, P. M. and Wills, J. M.) (Birkhäuser, Basel, 1983), 4996.
[12]Eggleston, H. G., Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, 47 (Cambridge University Press, New York, 1958).
[13]Everett, H., Stojmenovic, I., Valtr, P. and Whitesides, S., ‘The largest k-ball in a d-dimensional box’, Comput. Geom. 11(2) (1998), 5967.
[14]Favard, J., ‘Sur les corps convexes’, J. Math. Pures Appl. 12(9) (1933), 219282.
[15]Gluskin, E. D., ‘Norms of random matrices and diameters of finite-dimensional sets’, Mat. Sb. (N.S.) 120(162) (1983), 180189 (in Russian).
[16]Gritzmann, P. and Klee, V., ‘Inner and outer j-radii of convex bodies in finite-dimensional normed spaces’, Discrete Comput. Geom. 7 (1992), 255280.
[17]Henk, M., ‘Ungleichungen für sukzessive Minima und verallgemeinerte In- und Umkugelradien’, PhD Thesis, University of Siegen, 1991.
[18]Henk, M., ‘A generalization of Jung’s theorem’, Geom. Dedicata 42 (1992), 235240.
[19]Henk, M. and Hernández Cifre, M. A., ‘Intrinsic volumes and successive radii’, J. Math. Anal. Appl. 343(2) (2008), 733742.
[20]Hinrichs, A., ‘Approximation numbers of identity operators between symmetric Banach sequence spaces’, J. Approx. Theory 118 (2002), 305315.
[21]Hinrichs, A. and Michels, C., ‘Gelfand numbers of identity operators between symmetric sequence spaces’, Positivity 10 (2006), 111133.
[22]König, H., Eigenvalue Distributions of Compact Operators (Birkhäuser, Basel, 1986).
[23]Pietsch, A., ‘s-numbers of operators in Banach spaces’, Studia Math. 51 (1974), 201223.
[24]Pietsch, A., Operator Ideals (VEB Deutscher Verlag der Wissenschaften, Berlin, 1978).
[25]Pietsch, A., Eigenvalues and s-Numbers (Cambridge University Press, Cambridge, 1987).
[26]Pinkus, A., N-Widths in Approximation Theory (Springer, Berlin, 1985).
[27]Puhov, S. V., ‘Inequalities for the Kolmogorov and Bernšteĭn widths in Hilbert space’, Mat. Zametki 25(4) (1979), 619628; 637 (in Russian); translation Math. Notes 25 (4 (1979), 320–326.
[28]Sangwine-Yager, J. R., ‘Inner parallel bodies and geometric inequalities’, PhD Thesis, University of California Davis, 1978.
[29]Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, 2nd expanded edn (Cambridge University Press, Cambridge, 2014).
[30]Steckin, S. B., ‘On the best approximation of given classes of functions by arbitrary polynomials’, Uspekhi Mat. Nauk 9 (1954), 133–134 (in Russian).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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