Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T19:19:16.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Determination of Convex Bodies by Directed Projection Functions

Part of: General convexity

Published online by Cambridge University Press:  21 December 2009

Paul Goodey  and
Wolfgang Weil
Paul Goodey
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019, U.S.A. E-mail: [email protected]
Wolfgang Weil
Affiliation:
Mathematisches Institut II, Universität Karlsruhe, D-76128 Karlsruhe, Germany. E-mail: [email protected]
Get access

Abstract

A tensor-type integral formula for intrinsic volumes is used to define a further variant of directed projection functions and show that these determine a convex body uniquely. Averages of directed projection functions are then studied, and the connections between the resulting operators and previously considered spherical transforms discussed.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52A22: Random convex sets and integral geometry
Type
Research Article
Information
Mathematika , Volume 53 , Issue 1 , June 2006 , pp. 49 - 69
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

1Goodey, P., Kiderlen, M. and Weil, W., Section and projection means of convex bodies. Monatsh. Math. 126 (1998), 37–54.CrossRefGoogle Scholar
2Goodey, P. and Weil, W., Average section functions for star-shaped sets. Adv. in Appl. Math. 36 (2006), 70–84.CrossRefGoogle Scholar
3Goodey, P. and Weil, W., Directed projection functions of convex bodies. Monatsh. Math. 149 (2006), 43–64, 65 (Erratum).CrossRefGoogle Scholar
4Groemer, H., Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press (Cambridge, 1996).CrossRefGoogle Scholar
5Groemer, H., On the girth of convex bodies, Arch. Math. 69 (1997), 75–81.CrossRefGoogle Scholar
6Schneider, R., Zu einem Problem von Shephard über die Projektionen konvexer Körper. Math. Z. 101 (1967), 71–82.CrossRefGoogle Scholar
7Schneider, R., Functions on a sphere with vanishing integrals over certain subspheres. J. Math. Anal. Appl. 26 (1969), 381–384.CrossRefGoogle Scholar
8Schneider, R., Über eine Integralgleichung in der Theorie der konvexen Körper. Math. Nachr. 44 (1970), 55–75.CrossRefGoogle Scholar
9Schneider, R., Functional equations connected with rotations and their geometric applications. L'Enseignement Math. 16 (1970), 297–305.Google Scholar
10Schneider, R., Additive Transformationen konvexer Körper. Geom. Dedicata 3 (1974), 253–267.CrossRefGoogle Scholar
11Schneider, R., Convex Bodies: the Brunn-Minkowski Theory. Cambridge University Press (Cambridge, 1993).CrossRefGoogle Scholar
12Schneider, R., Tensor valuations on convex bodies and integral geometry. Rend. Circ. Mat. Palermo, Ser. II, Suppl. 65 (2000), 295–316.Google Scholar
13Schneider, R., Stable determination of convex bodies from projections, Monatsh. Math. (to appear) 2005.Google Scholar
14Schneider, R., Schuster, R., Tensor valuations on convex bodies and integral geometry, II. Rend. Circ. Mat. Palermo, Ser. II, Suppl. 70 (2002), 295–314.Google Scholar
15Weil, W., Translative and kinematic integral formulae for support functions. Geom. Dedicata 57 (1995), 91–103.CrossRefGoogle Scholar