Advertisement

Boundary structure and curvature of convex bodies

  • Rolf Schneider

Abstract

One of the fascinating features of the theory of convex bodies is the wealth of substantial results that spring from the mere assumption of convexity. The present survey is concerned with the implications that convexity of a point set has on the structure of its boundary. We have tried to collect the known results which describe, or are connected with, local geometric properties of the boundary of a convex body. Some open problems in this field will also be mentioned.

Keywords

Convex Body Convex Surface Curvature Measure Area Function Boundary Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [5]
    A.D. Aleksandrov, 1937a, Zur Theorie der gemischten Volumina von konvexen Körpern. I. Verallgemeinerung einiger Begriffe der Theorie der konvexen Körper. (Russian) Mat. Sbornik N.S. 2 (1937), 947–972.Google Scholar
  2. [9]
    A.D. Aleksandrov, 1937b, Zur Theorie der gemischten Volumina von konvexen Körpern. II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen. (Russian) Mat. Sbornik N.S. 2 (1937), 1205–1238.Google Scholar
  3. [10]
    A.D. Aleksandrov, 1937c, Über die Frage nach der Existenz eines konvexen Körpers, bei dem die Summe der Hauptkrümmungsradien eine gegebene positive Funktion ist, welche den Bedingungen der Geschlossenheit genügt. C.R. (Doklady) Acad. Sci. URSS 14 (1937), 59–60.Google Scholar
  4. [8]
    A.D. Aleksandrov, 1938, Zur Theorie der gemischten Volumina von konvexen Körpern. III. Die Erweiterung zweier Lehrsätze Minkowskis über die konvexen Polyeder auf beliebige konvexe Flächen. (Russian) Mat. Sbornik N.S. 3 (1938), 27–46.Google Scholar
  5. [10]
    A.D. Aleksandrov, 1939a, Anwendung des Satzes über die Invarianz des Gebietes auf Existenzbeweise. (Russian) Izv. Akad. Nauk SSSR 3 (1939), 243–256.Google Scholar
  6. [5]
    A.D. Aleksandrov, 1939b, Über die Oberflächenfunktion eines konvexen Körpers. (Russian) Mat. Sbornik N.S. 6 (48) (1939), 167–174.Google Scholar
  7. [4]
    A.D. Aleksandrov, 1939c, 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), 3–35.Google Scholar
  8. [9]
    A.D. Aleksandrov, 1942a, Existence and uniqueness of a convex surface with a given integral curvature. C.R. (Doklady) Acad. Sci. URSS 35 (1942), 131–134.Google Scholar
  9. [8]
    A.D. Aleksandrov, 1942b, Smoothness of the convex surface of bounded Gaussian curvature. C.R. (Doklady) Acad. Sci. URSS 36 (1942), 195–199.Google Scholar
  10. [1]
    A.D. Aleksandrov, 1948, Die innere Geometrie der konvexen Flächen. Akademie-Verlag, Berlin 1955 (Russian original: 1948 ).Google Scholar
  11. [10]
    A.D. Aleksandrov, 1950, Konvexe Polyeder. Akademie-Verlag, Berlin 1958 (Russian original: 1950 ).Google Scholar
  12. [9]
    A.D. Aleksandrov, 1961A, congruence condition for closed convex surfaces. (Russian) Vestnik Leningrad. Univ. 16 (1961), 5–7.Google Scholar
  13. [2]
    E.M. Alfsen, 1971, Compact convex sets and boundary integrals. Springer-Verlag, Berlin et al. 1971.Google Scholar
  14. [1]
    R.D. Anderson, V.L. Klee, 1952, Convex functions and upper semi-continuous collections. Duke Math. J. 19 (1952), 349 357. MR 13, p. 863.Google Scholar
  15. [2]
    E. Asplund, 1963, A k-extreme point is the limit of k-exposed points. Israel J. Math. 1 (1963), 161–162.Google Scholar
  16. [1]
    E. Asplund, 1968, Fréchet differentiability of convex functions. Acta Math. 121 (1968), 31–47.Google Scholar
  17. [4]
    E. Asplund, 1973, Differentiability of the metric projection in finite-dimensional Euclidean space. Proc. Amer. Math. Soc. 38 (1973), 218–219.Google Scholar
  18. [2]
    J. Bair, 1976, Extension du théorème de Straszewicz. Bull. Soc. roy. Sci. Liège 45 (1976), 166–168.Google Scholar
  19. [3]
    ]J. Bair, R. Fourneau, 1976, Etude géométrique des espaces vectoriels. II. Polyèdres et polytopes convexes. Mimeographed seminar notes, Université de Liège 1976.Google Scholar
  20. [10]
    I.Ja. Bakel’man, 1965, Geometric methods of solution of elliptic equations. (Russian) Izdat. “Nauka”, Moscow 1965.Google Scholar
  21. ]V. Bangert, 1977, Konvexität in riemannschen Mannigfaltigkeiten. Dissertation, Dortmund 1977.Google Scholar
  22. [0]
    V. Bangert, 1978a Konvexe Mengen in Riemannschen Mannigfaltigkeiten. Math. Z. 162 (1978), 263–286.Google Scholar
  23. [4]
    V. Bangert,1979, Konvexe Funktionen auf riemannschen Mannigfaltigkeiten. J. reine angew. Math. (to appear).Google Scholar
  24. [2]
    H. Bauer. 1964, Konvexität in topologischen Vektorräumen. Vorlesungsausarbeitung, Hamburg 1964.Google Scholar
  25. [9]
    Ch. Berg, 1969, Corps convexes et potentiels sphériques. Danske Vid. Selskab. Mat-fys. Medd. 37, 6 (1969).Google Scholar
  26. [1]
    A.S. Besicovitch, 1963a, On singular points of convex surfaces. Proc. Symp. Pure Math. 7 (Convexity), 21–23; Amer. Math. Soc., Providence 1963.Google Scholar
  27. [3]
    A.S. Besicovitch, 1963b, On the set of directions of linear segments on a convex surface. Proc. Symp. Pure Math. 7 (Convexity), 24–25; Amer. Math. Soc., Providence 1963.Google Scholar
  28. [1]
    W. Blaschke, 1916, Kreis und Kugel. Veit, Leipzig 1916. 2nd edition: de Gruyter, Berlin 1956.Google Scholar
  29. [9]
    R. Blind, 1977, Eine Charakterisierung der Sphäre im E3. manuscripta math. 21 (1977), 243–253.Google Scholar
  30. [7]
    J. Bokowski, H. Hadwiger, J.M. Wills, 1976 Eine Erweiterung der Croftonschen Formeln für konvexe Körper. Mathematika 23 (1976), 212–219.Google Scholar
  31. [1]
    T. Bonnesen, W. Fenchel, 1934, Theorie der konvexen Körper. Springer-Verlag, Berlin 1934.Google Scholar
  32. [1]
    T. Botts, 1942, Convex sets. Amer. Math. Monthly 49 (1942), 527–535. MR 4, p. 111.Google Scholar
  33. [0]
    Yu.D. Burago, V.A. Zalgaller, 1977, Convex sets in Riemannian spaces of non-negative curvature. (Russian) Uspekhi Mat. Nauk. 32: 3 (1977), 3–55. English translation: Russian Math. Surveys 32, 3 (1977), 1–57.Google Scholar
  34. [1]
    H. Busemann, 1958, Convex surfaces. Interscience Publishers, New York 1958.Google Scholar
  35. [9]
    H. Busemann, 1959, Minkowski’s and related problems for convex surfaces with boundaries. Michigan Math. J. 6 (1959), 259–266.Google Scholar
  36. [4]
    H. Busemann, W. Feller, 1935a, Bemerkungen zur Differentialgeometrie der konvexen Flächen. I. Kürzeste Linien auf differenzierbaren Flächen. Matematisk Tidsskrift B 1935, 25–36. FdM 61, p. 1429.Google Scholar
  37. [4]
    H. Busemann, W. Feller, 1935, Bemerkungen zur Differentialgeometrie der konvexen Flächen. II. Über die Krümmungsindikatrizen. Matematisk Tidsskrift B 1935, 87–115. FdM 61, p. 1429.Google Scholar
  38. [4]
    H. Busemann, W. Feller, 1936a, Krümmungseigenschaften konvexer Flächen. Acta Math. 66 (1936), 1–47. FdM 62, p. 832. Zbl 12, p. 274.Google Scholar
  39. [4]
    H. Busemann, W. Feller, 1936, Bemerkungen zur Differentialgeometrie der konvexen Flächen. III. Über die Gaussche Krümmung. Matematisk Tidsskrift B 1936, 41–70.Google Scholar
  40. [8]
    G.D. Chakerian, 1971, Higher dimensional analogues of an isoperimetric inequality of Benson. Math. Nachr. 48 (1971), 33–41. MR 44 # 4643. Zbl 214, p. 492.Google Scholar
  41. [2]
    G. Choquet, 1969, Lectures on analysis, vol. Il. W.A. Benjamin, New York et al. 1969.Google Scholar
  42. [2]
    G. Choquet, H. Corson, V. Klee, 1966, Exposed points of convex sets. Pacific J. Math. 17 (1966), 33 43.Google Scholar
  43. [2]
    J.B. Collier, 1975, On the set of extreme points of a convex body. Proc. Amer. Math. Soc. 47 (1975), 184–186.Google Scholar
  44. [3]
    J.B. Collier, 1976, On the facial structure of a convex body. Proc. Amer. Math. Soc. 61 (1976), 367–370.Google Scholar
  45. [2]
    H.H. Corson, 1965, A compact convex set in E3 whose exposed points are of the first category. Proc. Amer.Math. Soc. 16 (1965), 1015–1021.Google Scholar
  46. [9]
    V.I. Diskant, 1968, Stability of a sphere in the class of convex surfaces of bounded specific curvature. (Russian) Sibirskii Mat. Z. 9 (1968), 816–824. English translation: Siberian Math. J. 9 (1968), 610–615.Google Scholar
  47. [9]
    V.I. Diskant, 1971, Bounds for convex surfaces with bounded curvature functions. (Russian) Sibirskii Mat. 2. 12 (1971), 109–125. English translation: Siberian Math. J. 12 (1971), 78–89.Google Scholar
  48. [9]
    V.I. Diskant, 1972, Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference. (Russian) Sibirskii Mat. 2. 13 (1972), 767–772. English translation: Siberian Math. J. 13 (1972), 529–532.Google Scholar
  49. [2]
    H.G. Eggleston, B. Grünbaum, V. Klee, 1964, Some semicontinuity theorems for convex polytopes and cell complexes. Comment. Math. Helvet. 39 (1964), 165–188.Google Scholar
  50. [2]
    L.Q. Euler, 1977, Semi-continuity of the face-function for a convex set. Comment. Math. Helvet. 52 (1977), 325–328.Google Scholar
  51. [3]
    G. Ewald, 1964, Über die Schattengrenzen konvexer Körper. Abh. Math. Sem. Univ. Hamburg 27 (1964), 167–170.Google Scholar
  52. [3]
    G. Ewald, D.G. Larman, C.A. Rogers, 1970, 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), 1–20.Google Scholar
  53. [1]
    J. Favard, 1933a, Sur les corps convexes. J. Math. pures appl. (9) 12 (1933), 219–282.Google Scholar
  54. [8]
    J. Favard, 1933b, Sur la détermination des surfaces convexes. Bull. Acad. roy. Belgique (Cl. Sci.) 19 (1933), 65–75.Google Scholar
  55. [2]
    H. Federer, 1959, Curvature measures. Trans. Amer. Math. Soc. 93 (1958), 418–491.Google Scholar
  56. [4]
    H. Federer, 1969, Geometric measure theory. Springer-Verlag, Berlin, Heidelberg, New York 1969.Google Scholar
  57. [2]
    W. Fenchel, 1938, Über die neuere Entwicklung der Brunn-Minkowskischen Theorie der konvexen Körper. Congr. Math. Scand., Helsingfors 1938, 249–272.Google Scholar
  58. [9]
    W. Fenchel, B. Jessen, 1938, Mengenfunktionen und konvexe Körper. Danske Vid. Selskab. Mat.-fys. Medd. 16, 3 (1938).Google Scholar
  59. [10]
    W.J. Firey, 1965, The brightness of convex bodies. Technical Report no. 19, Oregon State University, 1965.Google Scholar
  60. [10]
    W.J. Firey, 1967a, Blaschke sums of convex bodies and mixed bodies. Proc. Colloquium Convexity (Copenhagen 1965), K9benhavns Univ. Mat. Inst. 1967, 94–101.Google Scholar
  61. [2]
    W.J. Firey, 1967b, The determination of convex bodies from their mean radius of curvature functions. Mathematika 14 (1967), 1–13.Google Scholar
  62. [10]
    W.J. Firey, 1967c, Generalized convex bodies of revolution. Canadian J. Math. 14 (1967), 972–996.Google Scholar
  63. [10]
    W.J. Firey, 1968, Christoffel’s problem for general convex bodies. Mathematika 15 (1968), 7–21.Google Scholar
  64. [10]
    W.J. Firey, 1970a, Local behaviour of area functions of convex bodies. Pacific J. Math. 35 (1970), 345–357.Google Scholar
  65. [10]
    W.J. Firey, 1970b, The determination of convex bodies by elementary symmetric functions of principal radii of curvature. Mimeographed manuscript, 1970.Google Scholar
  66. [10]
    W.J. Firey, 1970c Intermediate Christoffel-Minkowski problems for figures of revolution. Israel J. Math. 8 (1970), 384–390.Google Scholar
  67. [7]
    W.J. Firey, 1972, An integral-geometric meaning for lower order area functions of convex bodies. Mathematika 19 (1972), 205–212.Google Scholar
  68. [7]
    W.J. Firey, 1974, Kinematic measures for sets of support figures. Mathematika 21 (1974), 270–281.Google Scholar
  69. [7]
    W.J. Firey, 1975, Some open questions on convex surfaces. Proc. Int. Congr. Math., Vancouver 1974 (1975), 479–484.Google Scholar
  70. [8]
    W.J. Firey, 1979, Inner contact measures. Mathematika (to appear).Google Scholar
  71. [10]
    W.J. Firey, B. Grünbaum, 1964, Addition and decomposition of convex polytopes. Israel J. Math. 2 (1964), 91–100.Google Scholar
  72. [2]
    W.J. Firey, R. Schneider, 1979, The size of skeletons of convex bodies. Geometriae Dedicata 8 (1979), 99–103.Google Scholar
  73. [5]
    F.J. Flaherty, 1973, Curvature measures for piecewise linear manifolds. Bull. Amer. Math. Soc. 79 (1973), 100–102.Google Scholar
  74. [11]
    M. Fujiwara, 1916, Über die Anzahl der Kantenlinien einer geschlossenen konvexen Fläche. Tóhoku Math. J. 10 (1916), 164–166.Google Scholar
  75. [10]
    H. Gluck, 1975, Manifolds with preassigned curvature — a survey. Bull. Amer. Math. Soc. 81 (1975), 313–329.Google Scholar
  76. [10]
    D.M. Goikhman, 1974, The differentiability of volume in Blaschke lattices. (Russian) Sibirskii Mat. 2. 15 (1974), 1406–1408. English translation: Siberian Math. J. 15 (1974), 997–999.Google Scholar
  77. [10]
    P.R. Goodey, R. Schneider, 1979, On the intermediate area functions of convex bodies (submitted).Google Scholar
  78. [7]
    H. Groemer, 1979, Remarks on the average distance of convex sets (submitted).Google Scholar
  79. [4]
    Y.M. Gruber, 1977, Die meisten konvexen Körper sind glatt, aber nicht zu glatt. Math. Ann. 229 (1977), 259 266.Google Scholar
  80. [3]
    B. Grünbaum, 1967, Convex polytopes. Interscience Publishers, London et al. 1967.Google Scholar
  81. [7]
    H. Hadwiger, 1957, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag, Berlin,Göttingen, Heidelberg.Google Scholar
  82. [6]
    H. Hadwiger, 1975a, Eikörperrichtungsfunktionale und kinematische Integralformeln. Studienvorlesung Universität Bern 1975, mimeographed manuscript.Google Scholar
  83. [7]
    H. Hadwiger, 1975b, Eine Erweiterung der kinematischen Hauptformel der Integralgeometrie. Abh. Math. Sem. Univ. Hamburg 44 (1975), 84–90.Google Scholar
  84. [7]
    H. Hadwiger, R. Schneider, 1971, Vektorielle Integralgeometrie. Elem. Math. 26 (1971), 49–57.Google Scholar
  85. [7]
    J. Heuser, 1976, Ein neues kinematisches BerührmaB für konvexe Körper. Diplomarbeit, Freiburg 1976.Google Scholar
  86. [3]
    B.A. Ivanov, 1973, Über geradlinige Abschnitte auf dem Rand eines konvexen Körpers. (Russian) Ukrain. geom. Sbornik 13 (1973), 69–71. MR 51 # 13864. Zbl 286, p. 337.Google Scholar
  87. [3]
    H. Hadwiger, 1976, Exceptional directions for a convex body. (Russian) Mat. Zametki 20 (1976), 365–371. English translation: Math. Notes USSR 20 (1976), 763–766.Google Scholar
  88. [2]
    K. Jacobs, 1971, Extremalpunkte konvexer Mengen. Selecta Mathematica III, 90–118; Springer-Verlag,Berlin et al. 1971. Zbl 219, p. 282.Google Scholar
  89. [2]
    J.E. Jayne, C.A. Rogers, 1977, The extremal structure of convex sets. J. Functional Anal. 26 (1977), 251–288.Google Scholar
  90. [2]
    M. Jerison, 1954, A property of extreme points of compact convex sets. Proc. Amer. Math. Soc. 5 (1954), 782 783.Google Scholar
  91. [4]
    B. Jessen, 1929, Om konvekse Kurvers Krumning. Matematisk Tidsskrift B 1929, 50–62.Google Scholar
  92. [10]
    M. Kallay, 1975, The extreme bodies in the set of plane convex bodies with a given width function. Israel J.Math. 22 (1975), 203–207.Google Scholar
  93. [3]
    S. Karlin, L.S. Shapley, 1953, Geometry of moment spaces. Memoirs AMS 12. Amer. Math. Soc., Providence 1953.Google Scholar
  94. [2]
    V. Klee, 1955, A note on extreme points. Amer. Math. Monthly 62 (1955), 30–32.Google Scholar
  95. [3]
    V. Klee, 1957, Research problem No. 5. Bull. Amer. Math. Soc. 63 (1957), 419.Google Scholar
  96. [2]
    V. Klee, 1958, Extremal structure of convex sets. II. Math. Z. 69 (1958), 90–104.Google Scholar
  97. [3]
    V. Klee, 1969, Can the boundary of a d-dimensional convex body contain segments in all directions?Amer. Math. Monthly 76 (1969), 408–410.Google Scholar
  98. [2]
    V. Klee, M. Martin, 1970, Must a compact end set have measure zero? Amer. Math Monthly 77 (1970), 616–618.Google Scholar
  99. [3]
    V. Klee, M. Martin, 1971, Semi-continuity of the face-function of a convex set. Comment. Math. Helvet. 46 (1971), 1 12.Google Scholar
  100. [10]
    H. Kneser, W. Süss, 1932, Die Volumina in linearen Scharen konvexer Körper. Mat. Tidsskr. B 1932, 19–25.Google Scholar
  101. [4]
    J.B. Kruskal, 1969, Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping. Proc. Amer. Math. Soc. 23 (1969), 697–703.Google Scholar
  102. [10]
    S.S. Kutateladze, 1973, Blaschke structures in the programming of isoperimetric problems. (Russian) Mat. Zametki 14 (1973), 745–754. English translation: Math. Notes USSR 14 (1973), 985–989.Google Scholar
  103. [10]
    S.S. Kutateladze, 1976, Symmetry measures. (Russian) Mat. Zametki 19 (1976), 615–622. English translation: Math. Notes USSR 19 (1976), 372–375.Google Scholar
  104. [10]
    S.S. Kutateladze, A.M. Rubinov, 1969, Problems of isoperimetric type in a space of convex bodies. (Russian) Optimalnoje Planirovanie 14 (1969), 61–79.Google Scholar
  105. [3]
    D.G. Larman, 1971, On a conjecture of Klee and Martin for convex bodies. Proc. London Math. Soc. 23 (1971), 668–682.Google Scholar
  106. [2]
    D.G. Larman, 1977, On the one-skeleton of a compact convex set in a Banach space. Proc. London Math. Soc. 34 (1977), 117–144.Google Scholar
  107. [2]
    D.G. Larman, P. Mani, 1970, Gleichungen und Ungleichungen für die Gerüste von konvexen Polytopen und Zellenkomplexen. Comment. Math. Helvet. 45 (1970), 199–218.Google Scholar
  108. [3]
    D.G. Larman, C.A. Rogers, 1970, Paths in the one skeleton of a convex body. Mathematika 17 (1970), 293–314.Google Scholar
  109. [4]
    D.G. Larman, C.A. Rogers, 1971, Increasing paths on the one-skeleton of a convex body and the directions of line segments on the boundary of a convex body. Proc. London Math. Soc. 23 (1971), 683–698.Google Scholar
  110. [2]
    D.G. Larman, C.A. Rogers, 1973, The finite dimensional skeletons of a compact convex set. Bull. London Math. Soc. 5 (1973), 145–153.Google Scholar
  111. [3]
    K. Leichtweiß, 1978, Über einige Eindeutigkeitssätze für konvexe Körper. manuscripta math. 23 (1978), 213–245.Google Scholar
  112. [10]
    A.A. Magomedov, 1974, Konvexe Polyeder mit gegebenen Krümmungen mit Bedingungen. (Russian) Geometrijaitopologija, vyp. II. Editors I. Ja. Bakel’man et al., Leningradskii gosud. ped. institut im. A.I. Gercena, Leningrad 1974, 128–138.Google Scholar
  113. [1]
    J.T. Marti, 1977, Konvexe Analysis. Birkhäuser-Verlag, Basel et al. 1977.Google Scholar
  114. [5]
    G. Matheron, 1975, Random sets and integral geometry. Wiley, New York et al. 1975.Google Scholar
  115. [3]
    S. Mazur, 1933, Über konvexe Mengen in linearen normierten Räumen. Studia Math. 4 (1933), 70–84.FdM 49, p. 1074.Google Scholar
  116. [3]
    T.J. McMinn, 1960, On the line segments of a convex surface in E3. Pacific J. Math. 10 (1960), 943–946.Google Scholar
  117. [10]
    P. McMullen, 1973, Representations of polytopes and polyhedral sets. Geometriae Dedicata 2 (1973), 83–99.Google Scholar
  118. [7]
    P. McMullen, 1974a, A dice probability problem. Mathematika 21 (1974), 193–198.Google Scholar
  119. [4]
    P. McMullen, 1974b, On the inner parallel body of a convex body. Israel J. Math. 19 (1974), 217–219.Google Scholar
  120. [1]
    P. McMullen, G.C. Shephard, 1971, Convex polytopes and the Upper Bound Conjecture. Cambridge University Press 1971.Google Scholar
  121. [10]
    H. Minkowski, 1897, Allgemeine Lehrsätze über die konvexen Polyeder. Nachr. Ges. Wiss. Göttingen (1897), 198–219 (Ges. Abh., Berlin 1911 ).Google Scholar
  122. [10]
    H. Minkowski, 1903, Volumen und Oberfläche. Math. Ann. 57 (1903), 447–495 (Ges. Abh., Berlin 1911).FdM 34, p. 649.Google Scholar
  123. [2]
    H. Minkowski, 1911, Theorie der konvexen Körper, insbesondere Begründung ihres Oberflächenbegriffs. Ges.Abh., vol. II, 131–229; B.G. Teubner, Leipzig et al. 1911.Google Scholar
  124. [10]
    Z. Nâdeník, 1968, Erste Krümmungsfunktion der Rotationseiflächen. Gasopis Pést. Mat. 93 (1968), 127–133.Google Scholar
  125. [3]
    S. Papadopoulou, 1977, On the geometry of stable compact convex sets. Math. Ann. 229 (1977), 193–200.Google Scholar
  126. [2]
    R.R. Phelps, 1966, Lectures on Choquet’s theorem. D. van Nostrand, Princeton et al. 1966.Google Scholar
  127. [10]
    A.V. Pogorelov, 1960 Monge-Ampère equations of elliptic type. Noordhoff, Groningen 1964 (Russian original: 1960 ).Google Scholar
  128. [9]
    H. Minkowski, 1969, Extrinsic geometry of convex surfaces. Translations of Mathematical Monographs, vol. 35,Amer. Math. Soc., Providence, Rhode Island, 1973 (Russian original: 1969 ).Google Scholar
  129. [10]
    H. Minkowski, 1975, The Minkowski multidimensional problem. V.H. Winston and Sons, Washington D.C. 1978(Russian original: 1975 ).Google Scholar
  130. [2]
    G.B. Price, 1937, On the extreme points of convex sets. Duke Math. J. 3 (1937), 56–67.FdM 63, p. 668. Zbl 16, p. 229.Google Scholar
  131. [1]
    K. Reidemeister, 1921, Über die singulären Randpunkte eines konvexen Körpers. Math. Ann. 83 (1921), 116–118.Google Scholar
  132. [3]
    H.B. Reiter, N.M. Stavrakas, 1977, On the compactness of the hyperspace of faces. Pacific J. Math. 73 (1977), 193–196.Google Scholar
  133. [4]
    Ju.G. Resetnjak, 1968, Generalized derivatives and differentiability almost everywhere. (Russian) Mat. Sbornik 75 (117) (1968), 323–334. English translation: Math. USSR (Sbornik) 4 (1968), 293–302.Google Scholar
  134. [4]
    F. Riesz, B.Sz.-Nagy, 1956, Vorlesungen über Funktionalanalysis. Deutscher Verlag der Wissenschaften, Berlin 1956.Google Scholar
  135. [1]
    A.W. Roberts, D.E. Varberg, 1973, Convex functions. Academic Press, New York et al. 1973.Google Scholar
  136. [1]
    R.T. Rockafellar, 1970, Convex analysis. Princeton University Press 1970.Google Scholar
  137. [4]
    G.T. Sallee, 1972, Minkowski decomposition of convex sets. Israel J. Math. 12 (1972), 266–276.Google Scholar
  138. [2]
    Ju.A. akin, 1973, Convex sets, extreme points, and simplexes. (Russian) Itogi Nauki i Tekhniki (Mat. Analiz)11 (1973), 5–50; English translation: J. Soviet Math. 4 (1975), 625–655.Google Scholar
  139. [10]
    R. Schneider, 1967, Zu einem Problem von Shephard über die Projektionen konvexer Körper. Math. Z. 101 (1967), 71–82.Google Scholar
  140. [9]
    R. Schneider, 1970a, On the projections of a convex polytope. Pacific J. Math. 32 (1970), 799–803.Google Scholar
  141. [9]
    R. Schneider, 1970b, Über eine Integralgleichung in der Theorie der konvexen Körper. Math. Nachr. 44 (1970), 55–75.Google Scholar
  142. [1]
    R. Schneider, 1972a, Krümmungsschwerpunkte konvexer Körper (I). Abh. Math. Sem. Univ. Hamburg 37 (1972), 112–132.Google Scholar
  143. [5]
    R. Schneider, I972b, Krümmungsschwerpunkte konvexer Körper (II). Abh. Math. Sem. Univ. Hamburg 37 (1972), 204–217.Google Scholar
  144. [10]
    R. Schneider, 1974, On asymmetry classes of convex bodies. Mathematika 21 (1974), 12–18.Google Scholar
  145. [9]
    R. Schneider, 1975a, Remark on a conjectured characterization of the sphere. Ann. Polonici Math. 31 (1975), 187–190.Google Scholar
  146. [7]
    R. Schneider, 1975b, Kinematische Berührmaße für konvexe Körper. Abh. Math. Sem. Univ. Hamburg 44 (1975), 12–23.Google Scholar
  147. [7]
    R. Schneider, 1975c, Kinematische Berührmaße für konvexe Körper und Integralrelationen für Oberflächen¬maße. Math. Ann. 218 (1975), 253–267.Google Scholar
  148. [9]
    R. Schneider, 1976, Bestimmung eines konvexen Körpers durch gewisse Berührmaße. Arch. Math. 27 (1976), 99–105.Google Scholar
  149. [7]
    R. Schneider, 1977a, Eine kinematische Integralformel für konvexe Körper. Arch. Math. 28 (1977), 217–220.Google Scholar
  150. [5]
    R. Schneider, 1977b, Kritische Punkte und Krümmung für die Mengen des Konvexringes. L’Enseignement Math. 23 (1977).Google Scholar
  151. [10]
    R. Schneider, 1977c, Das Christoffel-Problem für Polytope. Geometriae Dedicata 6 (1977), 81–85.Google Scholar
  152. [9]
    R. Schneider, 1977d, Eine Charakterisierung der Kugel. Arch. Math. 29 (1977), 660–665.Google Scholar
  153. [9]
    R. Schneider, 1978a, Curvature measures of convex bodies. Ann. Mat. Pura Appl. 116 (1978), 101–134.Google Scholar
  154. [2]
    R. Schneider, 1978b, On the skeletons of convex bodies. Bull. London Math. Soc. 10 (1978), 84–85.Google Scholar
  155. [3]
    R. Schneider, 1978c, Kinematic measures for sets of colliding convex bodies. Mathematika 25 (1978), 1–12.Google Scholar
  156. [9]
    R. Schneider, 1978d, Über Tangentialkörper der Kugel. manuscripta math. 23 (1978), 269–278.Google Scholar
  157. [5]
    R. Schneider, 1979, Bestimmung konvexer Körper durch Krümmungsmaße. Comment. Math. Helvet. 54 (1979), 42–60.Google Scholar
  158. [5]
    R. Schneider, 1979a, Parallelmengen mit Vielfachheit und Steinerformeln. Geometriae Dedicata (to appear).Google Scholar
  159. [4]
    R. Schneider, 1979b, Nonparametric convex hypersurfaces with a curvature restriction (submitted).Google Scholar
  160. [9]
    R. Schneider, 1979c, On the curvatures of convex bodies. Math. Ann. 240 (1979), 177–181.Google Scholar
  161. [9]
    R. Schneider, W. Weil, 1970, Über die Bestimmung eines konvexen Körpers durch die Inhalte seiner Projektionen. Math. Z. 116 (1970), 338–348.Google Scholar
  162. [10]
    S.Z. Shefel,1977, Smoothness of the solution to the Minkowski problem. (Russian) Sibirskii Mat. Z. 18(1977), 472–475. English translation: Siberian Math. J. 18 (1977), 338–340.Zbl 357. 52004.Google Scholar
  163. [1]
    J. Stoer, Ch. Witzgall, 1970, Convexity and optimization in finite dimensions. I. Springer-Verlag, Berlin et al. 1970.Google Scholar
  164. [2]
    S. Straszewicz, 1935, Über exponierte Punkte abgeschlossener Punktmengen. Fund. Math. 24 (1935), 139–143.FdM 61, p. 756.Google Scholar
  165. [10]
    W. Süss, 1931, Bestimmung einer geschlossenen konvexen Fläche durch die Gaußsche Krümmung. S.-B.preuß. Akad. Wiss. 1931, 686–695.Google Scholar
  166. [10]
    W. Süss, 1932, Zusammensetzung von Eikörpern und homothetische Eiflächen. Töhoku Math. J. 35 (1932), 47–50.Google Scholar
  167. [10]
    W. Süss, 1933, Bestimmung einer geschlossenen konvexen Fläche durch die Summe ihrer Hauptkrümmungsradien. Math. Ann. 108 (1933), 143–148.Google Scholar
  168. [9]
    Ju A. Volkov, 1963, Stability of the solution of Minkowski’s problem. (Russian) Vestnik Leningrad.Univ., Ser.Mat. Meh. Astronom. 18 (1963), 33–43.Google Scholar
  169. [3]
    Z. Waksman, M. Epelman, 1976, On point classification in convex sets. Math. Scand. 38 (1976), 83–96.Google Scholar
  170. [0]
    R. Walter, 1979, Konvexität in Riemannschen Mannigfaltigkeiten. Jber. DMV (to appear).Google Scholar
  171. [2]
    H. Wegner, 1974, Two problems of measurability concerning convex sets in Euclidean spaces. Math. Ann. 211 (1974), 115–118.Google Scholar
  172. [4]
    W. Weil, 1973, Ein Approximationssatz für konvexe Körper. manuscripta math. 8 (1973), 335–362.Google Scholar
  173. [10]
    W. Weil, 1979a, On surface area measures of convex bodies. Geometriae Dedicata (to appear).Google Scholar
  174. [7]
    W. Weil, 1979b, Zufällige Berührung konvexer Körper durch q-dimensionale Ebenen (submitted).Google Scholar
  175. [7]
    W. Weil, 1979c, Berührwahrscheinlichkeiten für konvexe Körper (submitted).Google Scholar
  176. [7]
    W. Weil, 1979d, Kinematic integral formulas for convex bodies (see this volume).Google Scholar
  177. [2]
    R.J.-B. Wets, 1974, Über einen Satz von Klee und Straszewicz. Oper. Res. Verf. 19, VI. Oberwolfach-Tag. Oper.Res. 1973, 185–189 (1974).Google Scholar
  178. [3]
    Zapiski naucn. Sem.Leningrad. Otd. mat. Inst. Steklov 27 (1972), 67–72; English translation: J. Soviet Math. 3 (1975), 437–441.Google Scholar
  179. [4]
    T. Zamfirescu, 1979a, The curvature of most convex surfaces vanishes almost everywhere (submitted).Google Scholar
  180. [4]
    T. Zamfirescu, 1979b, Curvature properties of typical convex surfaces (in preparation).Google Scholar
  181. Further references (added in proof)Google Scholar
  182. [2]
    G.R. Burton, The measure of the s-skeleton of a convex body (to appear).Google Scholar
  183. [7]
    G.R. Burton, Subspaces which touch a Borel subset of a convex surface (to appear).Google Scholar
  184. [2]
    L. Dalla, D.G. Larman, Convex bodies with almost all k-dimensional sections polytopes (to appear).Google Scholar
  185. [3]
    G. Debs, Applications affines ouvertes et convexes compacts stables. Bull. Sc. math. (2)102 (1978), 401–414.Google Scholar
  186. ] S. Gallivan, On the number of strictly increasing paths in the one-skeleton of a convex body.Google Scholar
  187. [2]
    S. Gallivan, On the number of disjoint increasing paths in the one-skeleton of a convex body leading to a given exposed face. Israel J. Math. (to appear).Google Scholar
  188. [5]
    P.R. Goodey, Limits of intermediate surface area measures of convex bodies (to appear).Google Scholar
  189. [10]
    G. Matheron, Les érosions infinitésimales. Geometrical Probability and Biological Structures: Buffon’s 200th Anniversary. Edited by R.E. Miles and J. Serra. Lecture Notes in Biomathematics 23, Springer-Verlag, Berlin-Heidelberg-New York 1978, pp. 251–269.Google Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Rolf Schneider
    • 1
  1. 1.Mathematisches InstitutUniversität FreiburgFreiburgDeutschland

Personalised recommendations