Some New Elements of the Elliptic Brunn-Minkowski Theory

Article Sidebar

PDF
Published: Nov 12, 2018
Keywords:
Minkowski inequality, Brunn-Minkowski inequality, Kneser-Süss inequality, Minkowski’s determinant inequality, Blaschke summation, Matrix Blaschke summation, Mixed determinant, Matrix Kneser-Süss inequality

Main Article Content

Poramate Pranayanuntana
Department of Control Engineering, Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang (KMITL), Bangkok, Thailand

Abstract

In this paper, we present various matrix analog of notions and inequalities in convex geometry. We employ the well known notion of mixed determinant - and analog of the notion of mixed volume in convex geometry, and introduce the matrix version of Blaschke summation – an analog of the notion of Blaschke summation for convex bodies. With these notions we then can develop some matrix analogs of the convex geometry. In this paper, we also present one new inequality analog – the matrix version of Kneser-Süss inequality.


Keywords: Minkowski inequality, Brunn-Minkowski inequality, Kneser-Süss inequality, Minkowski’s determinant inequality, Blaschke summation, Matrix Blaschke summation, Mixed determinant, Matrix Kneser-Süss inequality


Corresponding author: E-mail: ppranaya@duke.poly.edu

Article Details

Issue
Vol. 5 No. 1 (2005)
Section
Original Research Articles

Copyright Transfer Statement

 The copyright of this article is transferred to Current Applied Science and Technology journal with effect if and when the article is accepted for publication. The copyright transfer covers the exclusive right to reproduce and distribute the article, including reprints, translations, photographic reproductions, electronic form (offline, online) or any other reproductions of similar nature.

 The author warrants that this contribution is original and that he/she has full power to make this grant. The author signs for and accepts responsibility for releasing this material on behalf of any and all co-authors.

Here is the link for download: Copyright transfer form.pdf

 

 

References

[1] Schneider, R., 1993. Conver Bodies: The Brunn-Minkowski Theory. Cambridge University Press, New York.
[2] Lutwak, E., April 1986. Volume of mixed bodies. Transactions of The American Mathematical Society, 294, 2: 487-500.
[3] Horn, R. A. and Johnson, C. R., 1985. Matrix Analysis. Cambridge University Press, New York.
[4] Marcus, M. and Minc, H., 1964 A Survey of Matrix Theory and Matrix Inequalities. Dover Publications, New York.
[5] Prasolov, V. V., 1994. Problems and Theorems in Linear Algebra, volume 134. American Mathematical Society, United States.
[6] Webster, R., 1994. Converity. Oxford University Press, New York.
[7] Brunn, H., 1887. Über Ovale und Eiflächen. PhD thesis, Dissertation, München.
[8] Brunn, H., 1889. Über Curven ohne Wendepunkte. Habilitationsschrift, München.
[9] Minkowski, H., 1910. Geometrie der Zahlen. Teubner, Leipzig.
[10] Brunn, H., 1894. Referat über eine Arbeit: Exacte Grundlagen für eine Theorie der Ovale. S.-B. Bayer. Akad. Wiss., pp. 93-111.