Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-09T12:55:17.556Z Has data issue: false hasContentIssue false
  • English
  • Français

Volume Inequalities and Additive Maps of Convex Bodies

Part of: General convexity

Published online by Cambridge University Press:  21 December 2009

Franz E. Schuster
Franz E. Schuster
Affiliation:
Forschungsgruppe Konvexe und Diskrete Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8–10/1046, A-1040 Wien, Austria. E-mail: [email protected]
Get access

Abstract

Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary, a new Brunn-Minkowski inequality is obtained for the volume of polar projection bodies.

MSC classification

Secondary: 52A39: Mixed volumes and related topics
Type
Research Article
Information
Mathematika , Volume 53 , Issue 2 , December 2006 , pp. 211 - 234
Copyright
Copyright © University College London 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bourgain, J. and Lindenstrauss, J., Projection bodies. In Geometric Aspects of Functional Analysis (1986/1987), Springer (Berlin, 1988), 250–270.CrossRefGoogle Scholar
2Bolker, E. D., A class of convex bodies. Trans. Amer. Math. Soc. 145 (1969), 323–345.CrossRefGoogle Scholar
3Bonnesen, T. and Fenchel, W., Theorie der konvexen Körper. Springer (Berlin, 1934).Google Scholar
4Busemann, H., Volume in terms of concurrent cross-sections. Pacific J. Math. 3 (1953), 1–12.CrossRefGoogle Scholar
5Dunkl, C. F., Operators and harmonic analysis on the sphere. Trans. Amer. Math. Soc. 125 (1966), 250–263.CrossRefGoogle Scholar
6Gardner, R. J., Intersection bodies and the Busemann-Petty problem. Trans. Amer. Math. Soc. 342 (1994), 435–445.CrossRefGoogle Scholar
7Gardner, R. J., A positive answer to the Busemann-Petty problem in three dimensions. Annals Math. (2) 140 (1994), 435–447.CrossRefGoogle Scholar
8Gardner, R. J., Geometric Tomography. Cambridge University Press (Cambridge, 1995).Google Scholar
9Gardner, R. J., Koldobsky, A. and Schlumprecht, T., An analytic solution to the Busemann-Petty problem on sections of convex bodies. Annals Math. (2) 149 (1999), 691–703.CrossRefGoogle Scholar
10Goodey, P. and Weil, W., The determination of convex bodies from the mean of random sections. Math. Proc. Camb. Phil. Soc. 112 (1992), 419–430.CrossRefGoogle Scholar
11Goodey, P. and Weil, W., Zonoids and generalizations. In Handbook of Convex Geometry (ed. Gruber, P. M. and Wills, J. M.), North-Holland (Amsterdam, 1993), 1297–1326.CrossRefGoogle Scholar
12Grinberg, E. and Zhang, G., Convolutions, transforms, and convex bodies. Proc. London Math. Soc. (31) 78 (1999), 77–115.CrossRefGoogle Scholar
13Hug, D. and Schneider, R., Stability results involving surface area measures of convex bodies. Rend. Circ. Mat. Palermo 70 (2002), 21–51.Google Scholar
14Kalton, N. J. and Koldobsky, A., Intersection bodies and Lp spaces. Advances Math. (to appear).Google Scholar
15Kiderlen, M., Blaschke- and Minkowski-endomorphisms of convex bodies. Trans. Amer. Math. Soc. (to appear).Google Scholar
16Koldobsky, A., Intersection bodies, positive definite distributions, and the Busemann-Petty problem. Amer. J. Math. 120 (1998), 827–840.CrossRefGoogle Scholar
17Koldobsky, A., A functional analytic approach to intersection bodies. Geom. Funct. Anal. 10 (2000), 1507–1526.CrossRefGoogle Scholar
18Leng, G. and Zhao, C., Inequalities for dual quermassintegrals of mixed intersection bodies. Proc. Indian Acad. Sci. 115 (2003), 79–91.Google Scholar
19Leng, G. and Zhao, C., Brunn-Minkowski inequality for mixed intersection bodies. J. Math. Anal. Appl. 301 (2005), 115–123.Google Scholar
20Ludwig, M., Projection bodies and valuations. Advances Math. 172 (2002), 158–168.CrossRefGoogle Scholar
21Ludwig, M., Minkowski valuations. Trans. Amer. Math. Soc. 357 (2005), 4191–4213.CrossRefGoogle Scholar
22Lutwak, E., Dual mixed volumes. Pacific J. Math. 58 (1975), 531–538.CrossRefGoogle Scholar
23Lutwak, E., Mixed projection inequalities. Trans. Amer. Math. Soc. 287 (1985), 91–105.CrossRefGoogle Scholar
24Lutwak, E., Volume of mixed bodies. Trans. Amer. Math. Soc. 294 (1986), 487–500.CrossRefGoogle Scholar
25Lutwak, E., Intersection bodies and dual mixed volumes. Advances Math. 71 (1988), 232–261.CrossRefGoogle Scholar
26Lutwak, E., Centroid bodies and dual mixed volumes. Proc. London Math. Soc. (3) 60 (1990), 365–391.CrossRefGoogle Scholar
27Lutwak, E., Inequalities for mixed projection bodies. Trans. Amer. Math. Soc. 339 (1993), 901–916.CrossRefGoogle Scholar
28Lutwak, E., Selected affine isoperimetric inequalities. In Handbook of Convex Geometry (ed. Gruber, P. M. and Wills, J. M.), North-Holland (Amsterdam, 1993), 151–176.CrossRefGoogle Scholar
29Milman, V. D. and Pajor, A., Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In Geometric Aspects of Functional Analysis (1987–88), Lecture Notes in Math. 1376, Springer (Berlin, 1989), 64–104.CrossRefGoogle Scholar
30Petty, C. M., Centroid surfaces. Pacific J. Math. 11 (1961), 1535–1547.CrossRefGoogle Scholar
31Petty, C. M., Isoperimetric problems. In Proc. Conf. on Convexity and Combinatorial Geometry, Univ. of Oklahoma, June 1971 (1972), 26–41.Google Scholar
32Schneider, R., Über eine Integralgleichung in der Theorie der konvexen Körper. Math. Nachr. 44 (1970), 55–75.CrossRefGoogle Scholar
33Schneider, R., Equivariant endomorphisms of the space of convex bodies. Trans. Amer. Math. Soc. 194 (1974), 53–78.CrossRefGoogle Scholar
34Schneider, R., Bewegungsäquivariante, additive und stetige Transformationen konvexer Bereiche. Arch. Math. 25 (1974), 303–312.CrossRefGoogle Scholar
35Schneider, R., Convex Bodies: the Brunn-Minkowski Theory. Cambridge University Press (Cambridge, 1993).CrossRefGoogle Scholar
36Schneider, R. and Weil, W., Zonoids and related topics. In Convexity and its Applications, Birkhäuser (Basel, 1983), 296–317.CrossRefGoogle Scholar
37Schuster, F. E., Convolutions and multiplier transformations of convex bodies. Trans. Amer. Math. Soc. (to appear).Google Scholar
38Zhang, G., Restricted chord projection and affine inequalities. Geom. Dedicata 53 (1999), 183–202.Google Scholar
39Zhang, G., Centred bodies and dual mixed volumes. Trans. Amer. Math. Soc. 345 (1994), 777–801.CrossRefGoogle Scholar
40Zhang, G., A positive solution to the Busemann-Petty problem in ℝ4. Annals Math. (2) 149 (1999), 535–543.CrossRefGoogle Scholar