Abstract
This paper is concerned with various geometric averages of sections or projections of convex bodies. In particular, we consider Minkowski and Blaschke sums of sections as well as Minkowski sums of projections. The main result is a Crofton-type formula for Blaschke sums of sections. This is used to establish connections between the different averages mentioned above. As a consequence, we obtain results which show that, in some circumstances, a convex body is determined by the averages of its sections or projections.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fallert H, Goodey P, Well W (1997) Spherical projections and centrally symmetric sets. Adv Math129: 301–322
Gardner R (1995) Geometric Tomography. Cambridge: Univ Press
Goodey P (1997) Radon transforms of projection functions. Math Proc Camb Phil Soc (to appear)
Goodey P (1997) Minkowski sums of projections of convex bodies. Mathematika (to appear)
Goodey P, Well W (1987) Translative integral formulae for convex bodies. Aequationes Math34: 64–77
Goodey P, Well W (1992) The determination of convex bodies from the mean of random sections. Math Proc Camb Phil Soc112: 419–430
Helgason S (1984) Groups and Geometric Analysis. Orlando: Academic Press
Müller C (1966) Spherical Harmonics. Berlin: Springer
Pohlmann S, Mecke J, Stoyan D (1981) Stereological formulas for stationary surface processes. Math Operations f Statist, Ser Statistics12: 429–440
Schneider R (1974) Additive Transformationen konvexer Körper. Geom Dedicata3: 221–228
Schneider R (1975) Kinematische Berührmaße für konvexe Körper und Integralrelationen für Oberflächenmaße. Math Ann218: 253–267
Schneider R (1977) Rekonstruktion eines konvexen Körpers aus seinen Projektionen. Math Nachr79: 325–329
Schneider R (1993) Convex Bodies: the Brunn-Minkowski Theory. Cambridge: Univ Press
Schneider R, Well W (1992) Integralgeometrie. Stuttgart: Teubner
Spriestersbach K (1997) Determination of a convex body from the average of projections and stability results. Math Proc Camb Phil Soc (to appear)
Stoyan D, Kendall WS, Mecke J (1995) Stochastic Geometry and Its Applications, 2nd edn. New York: Wiley
Well W (1982) Zonoide und verwandte Klassen konvexer Körper. Mh Math94: 73–84
Well W (1987) Point processes of cylinders, particles and flats. Acta Applicandae Math9: 103–136
Well W (1997) On the mean shape of particle processes. Adv Appl Prob (to appear)
Well W (1997) Mean bodies associated with random closed sets. Suppl Rend Circ Mat Palermo, Ser II50: 387–412
Well W (1997) The mean normal distribution of stationary random sets and particle processes. In:Jeulin D (ed) Advances in Theory and Applications of Random Sets, Proc Int Symposium, Fontainebleau, France, 9–11 Oct. 1996, Singapore: World Scientific Publ pp 21–33
Author information
Authors and Affiliations
Additional information
The research of the first author was supported in part by NSF grants DMS-9504249 and INT-9123373
Rights and permissions
About this article
Cite this article
Goodey, P., Kiderlen, M. & Weil, W. Section and projection means of convex bodies. Monatshefte für Mathematik 126, 37–54 (1998). https://doi.org/10.1007/BF01312454
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01312454