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A generalization of the matrix form of the Brunn-Minkowski inequality

Part of: General convexity

Published online by Cambridge University Press:  09 April 2009

Jun Yuan  and
Gangsong Lenga
Jun Yuan
Affiliation:
School of Mathematics and Computer ScienceNanjing Normal UniversityNanjing 210097 [email protected]
Gangsong Lenga
Affiliation:
Department of MathematicsShanghai UniversityShanghai 200444 P. [email protected]
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Abstract

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In this paper, we establish an extension of the matrix form of the Brunn-Minkowski inequality. As applications, we give generalizations on the metric addition inequality of Alexander.

2000 Mathematics subject classification: primary 52A40.

Keywords

Brunn-Minkowski inequalitymetric addition inequalityKy Fan inequalityBellman's inequalitymatrixsimplex

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 1 , August 2007 , pp. 125 - 134
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Alexander, R., The geometry of metric and linear space (Springer-Verlag, Berlin, 1975) pp. 57–65.CrossRefGoogle Scholar
[2]Bakelman, I. J., Convex analysis and nonlinear geometric elliptic equations (Springer, Berlin, 1994).CrossRefGoogle Scholar
[3]Beckenbach, E. F. and Bellman, R., Inequalities (Springer, Berlin, 1961).CrossRefGoogle Scholar
[4]Bergström, H., A triangle inequalityfor matrices (Den Elfte Skandiaviski Matematiker-kongress, Trondheim, 1949).Google Scholar
[5]Borell, C., ‘The Brunn-Minkowski inequality in Gauss space’, Invent Math. 30 (1975), 202–216.CrossRefGoogle Scholar
[6]Borell, C., ‘Capacitary inequality of the Brunn-Minkowski inequality type’, Math. Ann. 263 (1993), 179–184.CrossRefGoogle Scholar
[7]Burago, Y. D. and Zalgaller, V. A., Geometric inequalities, (Translated from Russian: Springer Series in Soviet Mathematics) (Springer, New York, 1988).CrossRefGoogle Scholar
[8]Fan, K., ‘Problem 4786’, Amer. Math. Monthly 65 (1958), 289.Google Scholar
[9]Fan, K., ‘Some inequalities concerning positive-definite Hermitian matrices’, Proc. Cambridge Phil. Soc. 51 (1958), 414–421.CrossRefGoogle Scholar
[10]Fiedler, M. and Markham, T., ‘Some results on the Bergström and Minkowski inequalities’, Linear Algebra Appl. 232 (1996), 199–211.CrossRefGoogle Scholar
[11]Gardner, R. J., Geometric tomography, Encyclopaedia of Mathematics and its Applications 58 (Cambridge University Press, Cambridge, 1995).Google Scholar
[12]Gardner, R. J., ‘The Brunn-Minkowski inequality’, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 355–405.CrossRefGoogle Scholar
[13]Gardner, R. J. and Gronchi, P., ‘A Brunn-Minkowski inequality for the integer lattice’, Trans. Amer. Math. Soc. 353 (2001), 3995–4024.CrossRefGoogle Scholar
[14]Haynesworth, E. V., ‘Note on bounds for certain determinants’, Duke Math. J. 24 (1957), 313–320.Google Scholar
[15]Haynesworth, E. V. ‘Bounds for determinants with positive diagonals’, Trans. Amer. Math. Soc. 96 (1960), 395–413.CrossRefGoogle Scholar
[16]Haynsworth, E. V., ‘Applications of an inequality for the Schur complement’, Proc. Amer. Math. Soc. 24 (1970), 512–516.CrossRefGoogle Scholar
[17]Horn, R. and Johnson, C. R., Matrix analysis (Cambridge University Press, Cambridge, 1985).CrossRefGoogle Scholar
[18]Leng, G. S., The Brunn-Minkowski inequality for volume differences’, Adv. in Appl. Math. 32 (2004), 615–624.CrossRefGoogle Scholar
[19]Li, C. K. and Mathias, R., ‘Extremal characterizations of Schur complement and resulting inequalities’, SIAM Rev. 42 (2000), 233–246.CrossRefGoogle Scholar
[20]Oppenheim, A., ‘Advanced problems 5092’, Amer. Math. Monthly 701 (1963), 444.CrossRefGoogle Scholar
[21]Osserman, R., ‘The Brunn-Minkowski inequality for multiplictities’, Invent. Math. 125 (1996), 405–411.Google Scholar
[22]Schneider, R., Convex bodies: The Brunn-Minkowski theory (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
[23]Yang, L. and Zhang, J. Z., ‘On Alexander's conjecture’, Chinese Sci. Bull. 27 (1982), 1–3.Google Scholar