Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 15
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Guo, LuJun and Leng, Gangsong 2015. VOLUME INEQUALITIES FOR THE Lp-SINE TRANSFORM OF ISOTROPIC MEASURES. Bulletin of the Korean Mathematical Society, Vol. 52, Issue. 3, p. 837.

    Goodey, Paul and Weil, Wolfgang 2012. A uniqueness result for mean section bodies. Advances in Mathematics, Vol. 229, Issue. 1, p. 596.

    Goodey, Paul and Weil, Wolfgang 2011. Local properties of intertwining operators on the sphere. Advances in Mathematics, Vol. 227, Issue. 3, p. 1144.

    Schuster, Franz E. 2008. Valuations and Busemann–Petty type problems. Advances in Mathematics, Vol. 219, Issue. 1, p. 344.

    Goodey, Paul and Weil, Wolfgang 2006. Average section functions for star-shaped sets. Advances in Applied Mathematics, Vol. 36, Issue. 1, p. 70.

    Goodey, Paul and Weil, Wolfgang 2006. Directed Projection Functions of Convex Bodies. Monatshefte für Mathematik, Vol. 149, Issue. 1, p. 43.

    Schuster, Franz E. 2006. Volume Inequalities and Additive Maps of Convex Bodies. Mathematika, Vol. 53, Issue. 02, p. 211.

    Schneider, Rolf 2001. On the mean normal measures of a particle process. Advances in Applied Probability, Vol. 33, Issue. 01, p. 25.

    Weil, Wolfgang 2001. Mixed Measures and Functionals of Translative Integral Geometry. Mathematische Nachrichten, Vol. 223, Issue. 1, p. 161.

    Goodey, Paul 1998. Minkowski sums of projections of convex bodies. Mathematika, Vol. 45, Issue. 02, p. 253.

    Weil, Wolfgang 1997. On the mean shape of particle processes. Advances in Applied Probability, Vol. 29, Issue. 04, p. 890.

    Schneider, Rolf 1996. Simple valuations on convex bodies. Mathematika, Vol. 43, Issue. 01, p. 32.

    Cabo, A. J. and Baddeley, A. J. 1995. Line transects, covariance functions and set convergence. Advances in Applied Probability, Vol. 27, Issue. 03, p. 585.

    Weil, Wolfgang 1995. The Estimation of mean shape and mean particle number in overlapping particle systems in the plane. Advances in Applied Probability, Vol. 27, Issue. 01, p. 102.

    GOODEY, Paul and WEIL, Wolfgang 1993. Handbook of Convex Geometry.

  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 112, Issue 2
  • September 1992, pp. 419-430

The determination of convex bodies from the mean of random sections

  • Paul Goodey (a1) and Wolfgang Weil (a2)
  • DOI:
  • Published online: 24 October 2008

Random sectioning of particles (compact sets in ℝ3 with interior points) is a familiar procedure in stereology where it is used to estimate particle quantities like volume or surface area from planar or linear sections (see, for example, the survey [23] or the book [20]). In the following, we study the problem whether the whole shape of a convex particle K can be estimated from random sections. If E is an IUR (isotropic, uniform, random) line or plane intersecting K then the intersection Xk = KE is a (k-dimensional, k = 1 or 2) random set. It is clear that the distribution of Xk determines K uniquely and that if E1,…, En are such flats, the most natural estimator for K would be the convex hull

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[3]T. Bonnesen and W. Fenchel . Theorie der Konvexen Körper (Springer-Verlag, 1934).

[5]P. Goodey and W. Weil . Translative integral formulae for convex bodies. Aequationes Math. 34 (1987), 6477.

[6]P. Goodey and W. Weil . Integral geometric formulae for projection functions. Geom. Dedicata 41 (1992), 117126.

[7]J. F. C. Kingman . Random secants of a convex body. J. Appl. Probab. 6 (1969), 660672.

[8]K. Leichtweiβ . Konvexe Mengen (Springer-Verlag, 1980).

[10]C. Mallows and J. Clark . Linear-intercept distributions do not characterize plane sets. J. Appl. Probab. 7 (1970), 240244.

[12]C. Müller . Spherical Harmonics (Springer-Verlag, 1966).

[14]R. Schneider . Zu einem Problem von Shephard über die Projektionen konvexer Körper. Math. Z. 101 (1967), 7182.

[15]R. Schneider . Über eine Integralgleichung in der Theorie der konvexen Körper. Math. Nachr. 44 (1970), 5575.

[16]R. Schneider . Rekonstruktion eines konvexen Körpers aus seinen Projektionen. Math. Nachr. 79 (1977), 325329.

[18]R. Schneider and W. Weil . Zonoids and related topics. In Convexity and its Applications (editors P. Gruber and J. M. Wills ) (Birkhäuser, 1983). pp. 296317.

[19]R. Schneider and W. Weil . Translative and kinematic integral formulae for curvature measures. Math. Nachr. 129 (1986), 6780.

[21]P. Waksman . Plane polygons and a conjecture of Blaschke's. Adv. in Appl. Probab. 17 (1985), 774793.

[22]W. Weil . Centrally symmetric convex bodies and distributions, II. Israel J. Math. 32 (1979), 173182.

[23]W. Weil . Stereology – a survey for geometers. In Convexity and its Applications (editors P. Gruber and J. M. Wills ) (Birkhäuser, 1983). pp. 360412.

[24]W. Weil . Iterations of translative integral formulae and non-isotropic Poisson processes of particles. Math. Z. 205 (1990), 531551.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *