J. Aust. Math. Soc. 83 (2007), no. 1, pp. 125–134.

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

Jun Yuan Gangsong Leng
School of Mathematics and Computer Science
Nanjing Normal University
Nanjing 210097
P.R. China
yuanjun@graduate.shu.edu.cn
Department of Mathematics
Shanghai University
Shanghai 200444
P. R. China
gleng@staff.shu.edu.cn
Received 4 March 2005; revised 24 March 2006
Communicated by A. Rubinov
Supported in part by the National Natural Science Foundation of China (Grant No. 10671117).

Abstract

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.

Download the article in PDF format (size 116 Kb)

2000 Mathematics Subject Classification: primary 52A40
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2354??? Z'blatt-MATH: pre05231336
indicates author for correspondence

References

  1. R. Alexander, The geometry of metric and linear space (Springer-Verlag, Berlin, 1975) pp. 57–65. MR405245
  2. I. J. Bakelman, Convex analysis and nonlinear geometric elliptic equations (Springer, Berlin, 1994). MR1305147
  3. E. F. Beckenbach and R. Bellman, Inequalities (Springer, Berlin, 1961). MR158038
  4. H. Bergström, A triangle inequality for matrices (Den Elfte Skandiaviski Matematiker-kongress, Trondheim, 1949).
  5. C. Borell, ‘The Brunn-Minkowski inequality in Gauss space’, Invent Math. 30 (1975), 202–216. MR399402
  6. C. Borell, ‘Capacitary inequality of the Brunn-Minkowski inequality type’, Math. Ann. 263 (1993), 179–184. MR698001
  7. Y. D. Burago and V. A. Zalgaller, Geometric inequalities, (Translated from Russian: Springer Series in Soviet Mathematics) (Springer, New York, 1988). MR936419
  8. K. Fan, ‘Problem 4786’, Amer. Math. Monthly 65 (1958), 289.
  9. K. Fan, ‘Some inequalities concerning positive-definite Hermitian matrices’, Proc. Cambridge Phil. Soc. 51 (1958), 414–421. MR76724
  10. M. Fiedler and T. Markham, ‘Some results on the Bergström and Minkowski inequalities’, Linear Algebra Appl. 232 (1996), 199–211. MR1366585
  11. R. J. Gardner, Geometric tomography, EncyclopŁdia of Mathematics and its Applications 58 (Cambridge University Press, Cambridge, 1995). MR1356221
  12. R. J. Gardner, ‘The Brunn-Minkowski inequality’, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 355–405. MR1898210
  13. R. J. Gardner and P. Gronchi, ‘A Brunn-Minkowski inequality for the integer lattice’, Trans. Amer. Math. Soc. 353 (2001), 3995–4024. MR1837217
  14. E. V. Haynesworth, ‘Note on bounds for certain determinants’, Duke Math. J. 24 (1957), 313–320. MR89176
  15. E. V. Haynesworth, ‘Bounds for determinants with positive diagonals’, Trans. Amer. Math. Soc. 96 (1960), 395–413. MR120242
  16. E. V. Haynsworth, ‘Applications of an inequality for the Schur complement’, Proc. Amer. Math. Soc. 24 (1970), 512–516. MR255580
  17. R. Horn and C. R. Johnson, Matrix analysis (Cambridge University Press, Cambridge, 1985). MR832183
  18. G. S. Leng, ‘The Brunn-Minkowski inequality for volume differences’, Adv. in Appl. Math. 32 (2004), 615–624. MR2042686
  19. C. K. Li and R. Mathias, ‘Extremal characterizations of Schur complement and resulting inequalities’, SIAM Rev. 42 (2000), 233–246. MR1778353
  20. A. Oppenheim, ‘Advanced problems 5092’, Amer. Math. Monthly 701 (1963), 444.
  21. R. Osserman, ‘The Brunn-Minkowski inequality for multiplictities’, Invent. Math. 125 (1996), 405–411. MR1400312
  22. R. Schneider, Convex bodies: The Brunn-Minkowski theory (Cambridge University Press, Cambridge, 1993). MR1216521
  23. L. Yang and J. Z. Zhang, ‘On Alexander's conjecture’, Chinese Sci. Bull. 27 (1982), 1–3. MR732034
Australian Mathematical Publishing Association Inc.

Valid XHTML 1.0 Transitional

Feedback