Abstract
In this paper, we establish two theorems for the quermassintegrals of convex bodies, which are the generalizations of the well-known Aleksandrov’ s projection theorem and Loomis-Whitney’ s inequality, respectively. Applying these two theorems, we obtain a number of inequalities for the volumes of projections of convex bodies. Besides, we introduce the concept of the perturbation element of a convex body, and prove an extreme property of it.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ball, K., Shadows of convex bodies, Trans. Amer. Math. Soc., 1991, 327: 891–901.
Lutwak, E., Mixed projection inequalities, Trans. Amer. Math. Soc., 1985, 287: 92–106.
Bourgain, J., Lindenstrauss, J.. Projection bodies, Israel Seminar (G. A. F. A) 1986-1987, Lecture Notes in Math. Vol. 1317, Berlin-New York: Springer-Verlag. 1988, 250–269.
Chakerian, G. D., Lutwak, E., Bodies with similar projections, Trans. Amer. Math. Soc., 1997, 349: 1811–1820.
Schneider, R., Weil, W., Zonoids and related topics, Convexity and its Applications (eds. Gruber, P. M., Wills, J. M.), Basel: Birkhäuser, 1983, 296–316.
Schneider, R., Convex Bodies: the Brunn- Minkowski Theory, Cambridge : Cambridge University Press, 1993.
Schneider, R., On the determination of convex bodies by projection and girth functions, Result Math., 1998, 33: 155–160.
Thompson, A. C., Minkowski Geometry, Cambridge: Cambridge University Press, 1996.
Petty, C. M., Projection bodies, in Proceedings, Coll Convexity, Copenhagen, 1965, Kφbenhavns Univ. Mat. Inst., 1967, 234–241.
Schneider, R., Zu einem problem von Shephard über die projectionen konvexer körper, Math. Z., 1967, 101: 71–81.
Ball, K., Volume ratios and a reverse isoprimetric inequalitity, J. London Math. Soc., 1991, 44(2): 351–359.
Gardner, R. J., Intersection bodies and the Busemann-Petty problem, Trans. Amer. Math. Soc., 1994, 342: 435–445.
Gardner, R. J., A positive answer to the Busemann-petty problem in three dimensions, Annals of Math., 1994, 140: 435–447.
Grinberg, E. L., Isoperimetric inequalities and identities fork-dimensional cross-sections of convex bodies, Math. Ann., 1991, 291: 75–86.
Goodey, P., Schneider, R., Weil, W., On the determination of convex bodies by projection functions, Bull. London Math. Soc., 1997, 29: 82–88.
Lutwak, E., Intersection bodies and dual mixed volumes, Adv. Math., 1988, 71: 232–261.
Zhang, G., Centered bodies and dual mixed volumes, Trans. Amer. Soc., 1994, 345: 777–801.
Zhang, G., Dual Kinematic formulas, Trans. Amer. Soc., 1999, 351: 985–995.
Ren, D. L., An Introduction to Integral Geometry (in Chinese), Shanghai: Science and Technology Press, 1988.
Burago, Y. D., Zalgaller, V. A., Geometric Inequalities, Heidelberg: Springer-Verlag, 1988.
Brascamp, H. J., Lieb, H. J., Best constants in Young’s inequality, its converse and Its generalization to more than three functions, Adv. Math., 1976, 20: 151–173.
Kawashima, T., Polytopes which are orthogonal projections of regular simplexes, Geom. Dedicata, 1991, 38: 73–85.
Yang, L., Zhang, J. Z., Pseudo-symmetric point set and geometric inequalities, Acta Math. Sinica, 1986, 6(29): 802–806.
Zhang, J. Z., Yang, L., The equispaced embedding of finite point set in pseudo-Euclidean space, Acta Math. Sinica, 1981, 4(24): 481–487.
Leichtweiβ, K., Konvexe Mengen, Berlin: Springer-Verlag. 1980.
Lutwak, E., Centroid bodies and dual mixed volumes, Proc. London Math. Soc., 1990, 60(3): 365–391.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Leng, G., Zhang, L. Extreme properties of quermassintegrals of convex bodies. Sci. China Ser. A-Math. 44, 837–845 (2001). https://doi.org/10.1007/BF02880133
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02880133