Dual Quermassintegral Differences for Intersection Body

Abstract

In this paper, we introduce the concept of dual quermassintegral differences. Further, we give the dual Brunn-Minkowski inequality and dual Minkowski inequality for dual quermassintegral differences for mixed intersection bodies.

Share and Cite:

Zhao, L. and Yuan, J. (2014) Dual Quermassintegral Differences for Intersection Body. Advances in Pure Mathematics, 4, 529-534. doi: 10.4236/apm.2014.410061.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Bonnesen, T. and Fenchel, W. (1987) Theory of Convex Bodies, BCS Associates, Moscow, ID; German Original: Springer, Berlin, 1934.
[2] Lutwak, E. (1988) Intersection Bodies and Dual Mixed Volumes. Advances in Mathematics, 71, 232-261.
http://dx.doi.org/10.1016/0001-8708(88)90077-1
[3] Gardner, R.J. (1994) Intersection Bodies and the Busemann-Petty Problem. Transactions of the American Mathematical Society, 342, 435-445. http://dx.doi.org/10.1090/S0002-9947-1994-1201126-7
[4] Gardner, R.J. (1994) A Positive Answer to the Busemann-Petty Problem in Three Dimensions. Annals of Mathematics, 140, 435-447. http://dx.doi.org/10.2307/2118606
[5] Goodey, P., Lutwak, E. and Weil, W. (1996) Functional Analytic Characterizations of Classes of Convex Bodies. Mathematische Zeitschrift, 222, 363-381. http://dx.doi.org/10.1007/BF02621871
[6] Zhang, G.Y. (1999) A Positive Solution to the Busemann-Petty Problem in . Annals of Mathematics, 149 535-543.
http://dx.doi.org/10.2307/120974
[7] Lutwak, E. (1993) Inequalities for Mixed Projection Bodies. Transactions of the American Mathematical Society, 339, 901-916. http://dx.doi.org/10.1090/S0002-9947-1993-1124171-8
[8] Zhao, C.J. and Cheung, W.S. (2003) On P-Quermassintegral Differences Function. Proceedings of the Indian Academy of Science, 116, 221-231.
[9] Zhao, C.J. and Leng, G.S. (2005) Brunn-Minkowski Inequality for Mixed Intersection Bodies. Journal of Mathematical Analysis and Applications, 301, 115-123. http://dx.doi.org/10.1016/j.jmaa.2004.07.013
[10] Gardner, R.J. (2006) Geometric Tomography. 2nd Edition, Cambridge University Press, New York.
http://dx.doi.org/10.1017/CBO9781107341029
[11] Lutwak, E. (1986) Volume of Mixed Bodies. Transactions of the American Mathematical Society, 294, 487-450.
http://dx.doi.org/10.1090/S0002-9947-1986-0825717-3
[12] Thompson, A.C. (1996) Minkowski Geometry. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9781107325845
[13] Leng, G.S. (2004) The Brunn-Minkowski Inequality for Volume Differences. Advances in Applied Mathematics, 32, 615-624. http://dx.doi.org/10.1016/S0196-8858(03)00095-2
[14] Zhang, G.Y. (1994) Centered Bodies and Dual Mixed Volumes. Transactions of the American Mathematical Society, 345, 777-801.
[15] Losonczi, L. and Páles, Z. (1997) Inequalities for Indefinite Forms. Journal of Mathematical Analysis and Applications, 205, 148-156. http://dx.doi.org/10.1006/jmaa.1996.5188
[16] Beckenbach, E.F. and Bellman, R. (1961) Inequalities. Springer, Berlin.
http://dx.doi.org/10.1007/978-3-642-64971-4

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.