Annali di Matematica Pura ed Applicata (1923 -)

, Volume 116, Issue 1, pp 101–134 | Cite as

Curvature measures of convex bodies

  • Rolf Schneider


The curvature measures, introduced by Federer for the sets of positive reach, are investigated in the special case of convex bodies. This restriction yields additional results. Among them are:(5.1), an integral-geometric interpretation of the curvature measure of order m, showing that it measures, in a certain sense, the affine subspaces of codimension m+1 which touch the convex body;(6.1), an axiomatic characterization of the (linear combinations of) curvature measures similar to Hadwiger's characterization of the quermassintegrals of convex bodies;(8.1), the determination of the support of the curvature measure of order m, which turns out to be the closure of the m-skeleton of the convex body. Moreover we give, for the case of convex bodies, a new and comparatively short proof of an integral-geometric kinematic formula for curvature measures.


Linear Combination Convex Body Additional Result Short Proof Curvature Measure 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Aeppli, Einige Ähnlichkeits- und Symmetriesätze für differenzierbare Flächen im Raum, Comment. Math. Helvet.,33 (1959), pp. 174–195.MathSciNetCrossRefGoogle Scholar
  2. [2]
    A. D. Aleksandrov,Zur Theorie der gemischten Volumina von konvexen Körpern. — I:Verallgemeinerung einiger Begriffe der Theorie der konvexen Körper (Russian), Mat. Sbornik, N.S.,2 (1937), pp. 947–972.zbMATHGoogle Scholar
  3. [3]
    A. D. Aleksandrov,Zur Theorie der gemischten Volumina von konvexen Körpern. — II:Neue Ungleichungen zwischen gemischten Volumina und ihre Anwendungen (Russian), Mat. Sbornik, N.S.,2 (1938), pp. 1205–1238.zbMATHGoogle Scholar
  4. [4]
    A. D. Aleksandrov, Über die Oberflächenfunktion eines konvexen Körpers(Russian), Mat. Sbornik,6 (48) (1939), pp. 167–174.zbMATHGoogle Scholar
  5. [5]
    A. D. Aleksandrov,Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it (Russian), Uchenye Zapiski Leningrad. Gos. Univ., Math. Ser.,6 (1939), pp. 3–35.MathSciNetGoogle Scholar
  6. [6]
    A. D. Aleksandrov,Existence and uniqueness of a convex surface with a given integral curvature, Comptes Rendus (Doklady) Acad. Sci. URSS,35, No. 5 (1942), pp. 131–134.MathSciNetzbMATHGoogle Scholar
  7. [7]
    A. D. Aleksandrov,Die innere Geometrie der knovexen Flächen, Akademie-Verlag, Berlin (1955).Google Scholar
  8. [8]
    R. B. Ash,Measure, integration, and functional analysis, Academic Press, New York and London (1972).zbMATHGoogle Scholar
  9. [9]
    H. Bauer,Wahrscheinlichkeitstheorie und Grundzüge der Maßtheorie, 2nd ed., W. de Gruyter, Berlin and New York (1974).zbMATHGoogle Scholar
  10. [10]
    H. Busemann,Convex surfaces, Interscience Publishers, New York (1958).zbMATHGoogle Scholar
  11. [11]
    G. Ewald -D. G. Larman -C. A. Rogers,The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space, Mathematika,17 (1970), pp. 1–20.MathSciNetCrossRefGoogle Scholar
  12. [12]
    H. Federer,Curvature measures, Trans. Amer. Math. Soc.,93 (1959), pp. 418–491.MathSciNetCrossRefGoogle Scholar
  13. [13]
    H. Federer,Geometric measure theory, Springer, Berlin, Heidelberg, and New York (1969).zbMATHGoogle Scholar
  14. [14]
    W. Fenchel -B. Jessen, Mengenfunktionen und konvexe Körper, Danske Vid. Selsk. Mat.-Fys. Medd.,16 (3) (1938), pp. 1–31.zbMATHGoogle Scholar
  15. [15]
    W. J. Firey,An integral-geometric meaning for lower order area functions of convex bodies, Mathematika,19 (1972), pp. 205–212.MathSciNetCrossRefGoogle Scholar
  16. [16]
    W. J. Firey,Kinematic measures for sets of support figures, Mathematika,21 (1974), pp. 270–281.MathSciNetCrossRefGoogle Scholar
  17. [17]
    W. J. Firey - R. Schneider,The size of skeletons of convex bodies, Geometriae Dedicata (to appear).Google Scholar
  18. [18]
    H. Hadwiger, Über eine Mittelwertformel für Richtungsfunktionale im Vektorraum und einige Anwendungen, J. reine angew. Math.,185 (1943), pp. 241–252.MathSciNetzbMATHGoogle Scholar
  19. [19]
    H. Hadwiger,Altes und Neues über konvexe Körper, Birkhäuser-Verlag, Basel and Stuttgart (1955).CrossRefGoogle Scholar
  20. [20]
    H. Hadwiger,Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, Berlin, Göttingen, and Heidelberg (1957).CrossRefGoogle Scholar
  21. [21]
    H. Hadwiger, Eine Erweiterung der kinematischen Hauptformel der Integralgeometrie, Abh. Math. Sem. Univ. Hamburg,44 (1975), pp. 84–90.MathSciNetCrossRefGoogle Scholar
  22. [22]
    H. Hadwiger,Eikörperrichtungsfunktionale und kinematische Integralformeln, Universität Bern (1975) (Manuscript).Google Scholar
  23. [23]
    W. Maak,Differential- und Integralrechnung, 2nd ed., Vandenhoeck und Ruprecht, Göttingen (1960).zbMATHGoogle Scholar
  24. [24]
    V. A. Zalgaller,k-dimensional directions singular for a convex body F in Rn (Russian), Zapiski naučn. Sem. Leningrad. Otd. Mat. Inst. Steklov,27 (1972), pp. 67–72. English translation: J. Soviet Math.,3 (1975), pp. 437–441.Google Scholar
  25. [25]
    G. Matheron,Random sets and integral geometry, Wiley, New York et al. (1975).zbMATHGoogle Scholar
  26. [26]
    P. McMullen,Non-linear angle-sum relations for polyhedral cones and polytopes, Math. Proc. Camb. Phil. Soc.,78 (1975), pp. 247–261.MathSciNetCrossRefGoogle Scholar
  27. [27]
    P. McMullen -C. C. Shephard,Convex polytopes and the upper bound conjecture, London Math. Soc. Lecture Note Series 3, Cambridge Univ. Press, Cambridge (1971).zbMATHGoogle Scholar
  28. [28]
    J. Neveu,Mathematische Grundlagen der Wahrscheinlichkeitstheorie, R. Oldenbourg Verlag, München and Wien (1969).zbMATHGoogle Scholar
  29. [29]
    L. S. Pontrjagin,Topologische Gruppen, Teil 1, Teubner, Leipzig (1957).Google Scholar
  30. [30]
    R. Schneider, Über eine Integralgleichung in der Theorie der konvexen Körper, Math. Nachr.,44 (1970), pp. 55–75.MathSciNetCrossRefGoogle Scholar
  31. [31]
    R. Schneider, Kinematische Berührmaße für konvexe Körper, Abh. Math. Sem. Univ. Hamburg,44 (1975), pp. 12–23.MathSciNetCrossRefGoogle Scholar
  32. [32]
    R. Schneider, Kinematische Berührmaße für konvexe Körper und Integralrelationen für Oberflächenmaße, Math. Ann.,218 (1975), pp. 253–267.MathSciNetCrossRefGoogle Scholar
  33. [33]
    R. Schneider, Kritische Punkte und Krümmung für die Mengen des Konvexringes, L'Enseignement Math.,23 (1977), pp. 1–6.MathSciNetzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata (1923 -) 1978

Authors and Affiliations

  • Rolf Schneider
    • 1
  1. 1.Freiburg i.Br.Germany

Personalised recommendations