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Convex Bodies of Constant Width

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Convexity and Its Applications

Abstract

A spherical ball obviously has the property that it can be arbitrarily rotated between two fixed parallel planes without losing contact with either plane. It has been known for a long time, certainly since the time of Euler, that there are other convex bodies with the same property. Such bodies are called convex bodies of constant width. Other names that have also been used are ‘convex bodies of constant breadth’, ‘equiwide convex bodies’, ‘orbiforms’ and ‘spheroforms’ (in the two and three-dimensional case, respectively) and several more; the occasionally used German ‘Gleichdick’ being one of the most charming.

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References

  1. Alexander, R. Problem 15. The geometry of metric and linear spaces (Proc. Conf., Michigan State Univ., East Lansing, Mich., 1974) p. 240. Lecture Notes in Math. Vol. 490, Springer, Berlin 1975. MR 53, 6419.

    Google Scholar 

  2. Auerbach, H. Sur un problème de M. Ulam concernant l’équilibre des corps flottants. Studia Math. 7 (1938), 121–142. Zbl. 18, p. 175.

    Google Scholar 

  3. Beck, A., M.N. Bleicher Packing convex sets into similar sets. Acta Math. Acad. Sci. Hung. 22 (3–4) (1971/72), 283–303. MR 45, 7613.

    Google Scholar 

  4. Benson, R.V. Euclidean geometry and convexity. McGraw-Hill Co., New York-Toronto-London 1966. MR 35, 844.

    Google Scholar 

  5. Beretta, L., A. Maxia Insiemi convessi e orbiformi. Univ. Roma e Ist. Naz. Alta Mat. Rend. Mat. (5)1(1940), 1–64. MR 1, p. 264.

    Google Scholar 

  6. Beretta, L., A. Sui vertici di un’orbiforme e sulle cuspidi della sua curva media. Boll. Un. Mat. Ital. (2) 2 (1940) 216–220. MR 2, p. 261.

    Google Scholar 

  7. Berwald, L. Integralgeometrie. XXV. Über die Körper konstanter Helligkeit. (Bemerkungen zu der vorstehenden Abhandlung.). Math. Z. 42 (1937), 737–738. Zbl. 16, p. 374.

    Google Scholar 

  8. Besicovitch, A.S. Measures of asymmetry of convex curves (II): Curves of constant width. J. London Math. Soc. 26 (1951), 81–93. MR 12, p. 850.

    Google Scholar 

  9. Besicovitch, A.S. A problem on a circle. J. London Math. Soc. 36 (1961), 241–244. MR 23, Al285.

    Google Scholar 

  10. Besicovitch, A.S. Minimum area of a set of constant width. Proc. Sympos. Pure Math., Vol. VII, pp. 13–14. Amer. Math. Soc., Providence, R.I. 1963. MR 27, 1878.

    Google Scholar 

  11. Besicovitch, A.S. On semicircles inscribed into sets of constant width. Proc. Sympos. Pure Math., Vol. VII, pp. 15–18. Amer. Math. Soc., Providence, R.I., 1963. MR 27, 2912.

    Google Scholar 

  12. Blanc, E. Sur une généralisation des domaines d’épaisseur constante. C.R. Acad. Sci. Paris 219 (1944), 662–663. MR 7, p. 475.

    Google Scholar 

  13. Blaschke, W. Einige Bemerkungen über Kurven and Flächen von konstanter Breite. Ber. d. Verh. d. Sächs. Akad. Leipzig 67 (1915), 290–297. Jbuch. 45, p. 731.

    Google Scholar 

  14. Blaschke, W. Kreis und Kugel. Zweite Aufl., Walter de Gruyter Co., Berlin 1956. MR 14, p. 1123.

    Google Scholar 

  15. Blatter, C. Über Kurven konstanter Breite. Elem. Math. 36 (1981), 105–115.

    Google Scholar 

  16. Boltjanskii, V.G. The partitioning of plane figures into parts of smaller diameter. (Russian). Colloq.. Math. 21 (1970), 253–263. MR 42, 2366.

    Google Scholar 

  17. Boltyanskii, V.G., I.T. Gohberg The Decomposition of Figures into Smaller Parts. (Translated and adapted from the Russian by H. Christoffers and T.P. Branson). Univ. of Chicago Press, Chicago, Ill.-London, 1980. MR 81a:52001.

    Google Scholar 

  18. Bonnesen, T., W. Fenchel Theorie der konvexen Körper. Ergebn. d. Math. u. ihrer Grenzgeb. Bd. 3, J. Springer Verl., Berlin 1934. Zbl. 8, p. 77.

    Google Scholar 

  19. Bontrager, M.M. Characterizations and properties of sets of constant width. Ed. D. Dissertation, Oklahoma State University, Stillwater, 1969.

    Google Scholar 

  20. Bose, R.C., S.N. Roy Some properties of the convex oval with reference to its perimeter centroid. Bull. Calcutta Math. Soc. 27 (1935), 79–86. Zbl. 14, p. 127.

    Google Scholar 

  21. Buchman, E.O., F.A. Valentine Any new Helly numbers? Amer. Math. Montly 89 (1982), 370–375.

    Google Scholar 

  22. Bückner, H. Über Flächen von fester Breite. Jber. Deutsch. Math.-Verein. 46 (1936), 96–139. Zbl. 14, p. 361.

    Google Scholar 

  23. a Bückner, H. Ergänzungen zu meiner Arbeit: “Über Flächen von fester Breite”. (Dieser Jahresber. d. D.M.V. 46 Heft 5–8). Jber. Deutsch. Math.-Verein. 47 (1937), 55. Zbl. 16, p. 228.

    Google Scholar 

  24. b Bückner, H. Zur Differenzialgeometrie der Kurven und Flächen fester Breite. Schr. d. Königsberg. gel. Ges. Jg. 14, H.1; Halle a. d. S., Max Niemeyer 1937, 22 pp. Zbl. 17, p. 188.

    Google Scholar 

  25. Burke, J.F. A curve of constant diameter. Math. Mag. 39 (1966), 84–85. MR 33, 1794.

    Google Scholar 

  26. Chakerian, G.D. The affine image of a convex body of constant breadth. Israel J. Math. 3 (1965), 19–22. MR 32, 382.

    Google Scholar 

  27. Chakerian, G.D. Sets of constant width. Pacific J. Math. 19 (1966), 13–21. MR 34, 4986.

    Google Scholar 

  28. Chakerian, G.D. Sets of constant relative width and constant relative brightness. Trans. Amer. Math. Soc. 129 (1967), 26–37. MR 35, 3545.

    Google Scholar 

  29. Chakerian, G.D. Intersection and covering properties of convex sets. Amer. Math. Monthly 76 (1969), 753–766. MR 40, 3433.

    Google Scholar 

  30. Chakerian, G.D. A characterization of curves of constant width. Amer. Math. Monthly 81 (1974), 153–155. MR 49, 1321.

    Google Scholar 

  31. Chakerian, G.D. Mixed volumes and geometric inequalities. Convexity and Related Combinatorial Geometry. Proceedings of the Second University of Oklahoma Conference, pp. 57–62. Marcel Dekker, Inc., New York and Basel 1982.

    Google Scholar 

  32. Chakerian, G.D., M.S. Klamkin Minimal covers for closed curves. Math. Mag. 46 (1973), 55–61. MR 47, 2496.

    Google Scholar 

  33. Chakerian, G.D., G.T. Sallee An intersection theorem for sets of constant width. Duke Math. J. 36 (1969), 165–170. MR 39, 3392.

    Google Scholar 

  34. Chakerian, G.D., S.K. Stein Bisected chords of a convex body. Arch. Math. (Basel) 17 (1966), 561–565. MR 34, 6635.

    Google Scholar 

  35. Cooke, G. A problem in convexity. J. London Math. Soc. 39 (1964), 370–378. MR 29, 521.

    Google Scholar 

  36. Danzer, L. Über die maximale Dicke der ebenen Schnitte eines konvexen Körpers. Arch. Math. (Basel) 8 (1957), 314–316. MR 19, p. 1074.

    Google Scholar 

  37. Danzer, L. A characterization of the circle. Proc. Sympos. Pure Math., Vol. VII, pp. 99–100. Amer. Math. Soc., Providence, R.I. 1963. MR 27, 4141b.

    Google Scholar 

  38. Danzer, L., B. Grünbaum, V.Klee Helly’s theorem and its relatives. Proc. Sympos. Pure Math., Vol. VII, pp. 101–180. Amer. Math. Soc., Providence, R.I. 1963. MR 28, 524.

    Google Scholar 

  39. Debrunner, H. Zu einem massgeometrischen Satz über Körper konstanter Breite. Math. Nachr. 13 (1955), 165–167. MR 17, p. 294.

    Google Scholar 

  40. Dinghas, A. Verallgemeinerung eines Blaschkeschen Satzes über konvexe Körper konstanter Breite. Rev. Math. Union Interbalkan. 3 (1940), 17–20. MR 2, p. 261.

    Google Scholar 

  41. Dubois, J. Sur la convexité et ses applications. Ann. Sci. Math. Québec 1 (1977), no. 1, 7–31. MR 58, 24002.

    Google Scholar 

  42. Eggleston, H.G. Measures of asymmetry of convex curves of constant width and restricted radii of curvature. Quart. J. Math., Oxford Ser. (2) 3 (1952), 63–72. MR 13, p. 768.

    Google Scholar 

  43. Eggleston, H.G. A proof of Blaschke’s theorem on the Reuleaux triangle. Quart. J. Math., Oxford Ser. (2) 3 (1952), 296–297. MR 14, p. 496.

    Google Scholar 

  44. Eggleston, H.G. Sets of constant width contained in a set of given minimal width. Mathematika 2 (1955), 4855. MR 17, p. 185.

    Google Scholar 

  45. Eggleston, H.G. Covering a three-dimensional set with sets of smaller diameter. J. London Math. Soc. 30 (1955), 11–24. MR 16, p. 743.

    Google Scholar 

  46. Eggleston, H.G. Problems in Euclidean Space: Applications of convexity. International Series of Monographs on Pure and Applied Mathematics, Vol. V. Pergamon Press, New York-LondonParis-Los Angeles, 1957. MR 23, A3228.

    Google Scholar 

  47. Eggleston, H.G. Figures inscribed in convex sets. Amer. Math. Montly 65 (1958), 76–80. MR 20, 4235. 1958b Convexity. Cambridge Tracts in Mathematics and Mathematical Physics, No. 47. Cambridge Univ. Press, New York, 1959. MR 23, A2123.

    Google Scholar 

  48. Eggleston, H.G. Squares contained in plane convex sets. Quart. J. Math., Oxford Ser. (2) 10 (1959), 261–269. MR 22, 8436.

    Google Scholar 

  49. Eggleston, H.G. Minimal universal covers in E. Israel J. Math. 1 (1963), 149–155. MR 28, 4432:

    Google Scholar 

  50. Eggleston, H.G. Sets of constant width in finite dimensional Banach spaces. Israel J. Math. 3(1965), 163–172. MR 34, 583.

    Google Scholar 

  51. Eggleston, H.G., S.J. Taylor On the size of equilateral triangles which may be inscribed in curves of constant width. J. London Math. Soc. 27 (1952), 438–448. MR 14, 401.

    Google Scholar 

  52. Elkington, G.B., J. Hammer Lattice points in a convex set of given width. J. Austral. Math. Soc. Ser. A 21(1976), no. 4, 504–507. MR 54, 3594.

    Google Scholar 

  53. Ewald, G. Von Klassen konvexer Körper erzeugte Hilberträume. Math. Ann. 162 (1965/66), 140–146. MR 32, 8255.

    Google Scholar 

  54. Ewald, G., G.C. Shephard Normed vector spaces consisting of classes of convex sets. Math. Z. 91(1966), 1–19. MR 32, 4597.

    Google Scholar 

  55. Falconer, K.J. Applications of outwardly simple line families to plane convex sets. Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 3, 353–356. MR 56, 3744.

    Google Scholar 

  56. Falconer, K.J. A characterization of plane curves of constant width. J. London Math. Soc. (2) 16 (1977), no. 3, 536–538. MR 57, 1272.

    Google Scholar 

  57. Falconer, K.J. Singularities of sets of constant width. Geometriae Dedicata 11 (1981), 187–193.

    Google Scholar 

  58. Fejes Töth, G. New results in the theory of packing and covering. In convexity and its applications, ed. By Gruber and Wills, pp. 318–359 (Birkhäuser, Basel 1983).

    Google Scholar 

  59. Fejes Tóth, L. On the number of equal discs that can touch another of the same kind. Studia Sci. Math. Hungar. 2 (1967), 363–367. MR 36, 4440.

    Google Scholar 

  60. Fejes Tóth, L. Über Scheiben mit richtungsinvarianter Packungsdichte. Elem. Math. 26 (1971), 58–60. MR 45, 1045.

    Google Scholar 

  61. Fillmore, J.P. Symmetries of surfaces of constant width. J. Differential Geometry 3 (1969), 103–110. MR 40, 858.

    Google Scholar 

  62. Fillmore, J.P. Barbier’s theorem in the Lobachevski plane. Proc. Amer. Math. Soc. 24 (1970), 705–709. MR 40, 6365.

    Google Scholar 

  63. Firey, W.J. A note on a theorem of Knothe. Michigan Math. J. 6 (1959), 53–54. MR 21, 319.

    Google Scholar 

  64. Firey, W.J. Isoperimetric ratios of Reuleaux polygons. Pacific J. Math. 10 (1960), 823–829. MR 22, 4014.

    Google Scholar 

  65. Firey, W.J. Lower bounds for volumes of convex bodies. Arch. Math. (Basel) 16 (1965), 69–74. MR 31, 5152.

    Google Scholar 

  66. Firey, W.J. The brightness of convex bodies. Tech. Report No. 19, Oregon State Univ., Corvallis, Oregon 1965.

    Google Scholar 

  67. Firey, W.J. Blaschke sums of convex bodies and mixed bodies. Proc. Colloquium on Convexity (Copenhagen, 1965), pp. 94–101. Københavns Univ. Math. Inst., Copenhagen, 1967. MR 36, 3231.

    Google Scholar 

  68. Firey, W.J. Convex bodies with constant outer p-measure. Mathematika 17 (1970), 21–27. MR 42, 2367.

    Google Scholar 

  69. Firey, W.J., H. Groemer Convex polyhedra which cover a convex set. J. London Math. Soc. 39 (1964), 261–266, MR 29, 2707.

    Google Scholar 

  70. Firey, W.J., B. Grünbaum Addition and decomposition of convex polytopes, Israel J. Math. 2 (1964), 91–100. MR 30, 5218.

    Google Scholar 

  71. Fischer, H.J. Kurven, in denen ein Drei-oder Vieleck so herumbewegt werden kann, dass seine Ecken die Kurve durchlaufen. Deutsche Math. 1(1936), Zbl. 14, p. 409.

    Google Scholar 

  72. Florian, A. On the permeability of layers of discs. Studia Sci. Math. Hungar. 13 (1978), 125–132.

    Google Scholar 

  73. Florian, A. Über die Durchlässigkeit einer Schicht konvexer Scheiben. Studia Sci. Math. Hungar. 15 (1980), 201–213.

    Google Scholar 

  74. Focke, J. Symmetrische n-Orbiformen kleinsten Inhalts. Acta Math. Acad. Sci. Hungar. 20 (1969), 39–68. MR 38, 6463.

    Google Scholar 

  75. Focke, J. Die beste Ausbohrung eines regulären n-Ecks (English and Russian summaries). Z. Angew. Math. Mech. 49 (1969), 235–248. MR 39, 6167.

    Google Scholar 

  76. Focke, J., B. Gensel n-Orbiformen maximaler Randknickung. Beiträge Anal. Heft 2 (1971), 7–16. MR 46, 9862.

    Google Scholar 

  77. Freilich, G. On sets of constant width. Proc. Amer. Math. Soc. 2 (1951), 92–96. MR 12, p. 850.

    Google Scholar 

  78. Fujiwara, M. On space curves of constant breadth. Tôhoku Math. J. 5 (1914), 180–184. Jbuch. 45, p. 731.

    Google Scholar 

  79. Fujiwara, M. Über die einem Vielecke eingeschriebenen und umdrehbaren konvexen geschlossenen Kurven. Sci. Rep. Tôhoku Univ. 4 (1915), 43–55. Jbuch. 45, p. 1348.

    Google Scholar 

  80. Fujiwara, M. Über die Mittelkurve zweier geschlossener konvexer Kurven in bezug auf einen Punkt. Tôhoku Math. J. 10 (1916), 99–103. Jbuch. 46, p. 1117.

    Google Scholar 

  81. Gale, D. On inscribing n-dimensional sets in a regular n-simplex. Proc. Amer. Math. Soc. 4 (1953), 222–225. MR 14, p. 787.

    Google Scholar 

  82. Ganapathi, P. On a certain class of ovals. Math. Z. 38 (1934), 687–690. Zbl. 9, p. 124.

    Google Scholar 

  83. Ganapathi, P. The isoperimetric deficit of a convex closed curve. Math. Z. 38 (1934), 691–694. Zbl. 9, p. 124.

    Google Scholar 

  84. Ganapathi, P. A property of ovaloids of constant breadth. Tôhoku Math. J. 40 (1935), 421–424. Zbl. 11, p. 318.

    Google Scholar 

  85. Garay de Pablo, J., F. Clavera Emperador On polygons of constant affine width (Spanish). Rev. Acad. Ci. Zaragoza (2)19 (1964), 75–78. MR 31, 2661.

    Google Scholar 

  86. Garay de Pablo, J., F. Consideraciones acerca de los poligonos de anchura afin constante. Actas 5 reun. anual mat. esp., Valencia 1964, pp. 94–98. Publ. Inst. “Jorge Juan” Mat., Madrid, 1967. MR 39, 7503. Gardner, M.

    Google Scholar 

  87. Garay de Pablo, J., F. Mathematical games. Curves of constant width, one of which makes it possible to drill square holes. Scientific American, Feb. 1963, 148–156.

    Google Scholar 

  88. Gere, B.H., D. Zupnik On the construction of curves of constant width. J. Math. Phys. Mass. Inst. Tech. 22 (1943), 31–36. MR 5, p. 10.

    Google Scholar 

  89. Godbersen, C. Der Satz vom Vektorbereich in Räumen beliebiger Dimension. Diss. Göttingen 1938, 49 pp. Zbl. 20, p. 77.

    Google Scholar 

  90. Goldberg, M. Circular-arc rotors in regular polygons. Amer. Math. Monthly 55 (1948), 393–402. MR 10, 205.

    Google Scholar 

  91. Goldberg, M. Rotors in spherical polygons. J. Math. Physics 30 (1952), 235–244. MR 13, p. 577.

    Google Scholar 

  92. Goldberg, M. Rotors within rotors. Amer. Math. Monthly 61(1954), 166–171. MR 15, p. 740.

    Google Scholar 

  93. Goldberg, M. Basic rotors in spherical polygons. J. Math. Phys. 34 (1956), 322–327. MR 17, p. 655.

    Google Scholar 

  94. Goldberg, M. Trammel rotors in regular polygons. Amer. Math. Monthly 64 (1957), 71–78. MR 18, p. 668.

    Google Scholar 

  95. Goldberg, M. Rotors tangent to n fixed circles. J. Math. Phys. 37 (1958), 69–74. MR 20, 1438.

    Google Scholar 

  96. Goldberg, M. Intermittent rotors. J. Math. Phys. 38 (1959/60), 135–140. MR 21, 5936.

    Google Scholar 

  97. Goldberg, M. Rotors in polygons and polyhedra. Math. Comput. 14 (1960), 229–239. MR 22, 5934.

    Google Scholar 

  98. Goodey, P.R. Intersection of circles and curves of constant width. Math. Ann. 208 (1974), 49–58. MR 49, 6028.

    Google Scholar 

  99. Goodey, P.R. Intersections of convex bodies and surfaces. III. Geometry Symposium, Siegen, 14–17 July 1982. Mimeographed abstracts.

    Google Scholar 

  100. Goodey, P.R., M.M. Woodcock A characterization of sets of constant width. Math. Ann. 238 (1978), no. 1, 15–21. MR 80b:52007.

    Google Scholar 

  101. Gray, C.G. Solids of constant breadth. Math. Gaz. 56 (1972), no. 398, 289–292. MR 58, 7392.

    Google Scholar 

  102. Groemer, H. Eine Bemerkung über die Überdeckung des Raumes mit unsymmetrischen konvexen Körpern. Monatsh. Math. 66 (1962), 410–416. MR 30, 4192.

    Google Scholar 

  103. Groemer, H. Über die Quermassintegrale von konvexen Körpern konstanter Breite. (Unpublished).

    Google Scholar 

  104. Groemer, H. Extremal convex sets. To appear.

    Google Scholar 

  105. Gruber, P.M. Approximation of convex bodies. In Convexity and its applications, ed. by Gruber and Wills, pp. 131–162 (Birkhäuser, Basel 1983).

    Google Scholar 

  106. Gruber, P.M., R. Schneider Problems in geometric convexity. Contributions to Geometry (Proc. Geom. Sympos., Siegen 1978), pp. 255–278, Birkhäuser, Basel 1979. MR 82d:52001.

    Google Scholar 

  107. Grünbaum, B. Measures of symmetry for convex sets. Proc. Sympos. Pure Math., Vol. VII, pp. 233–270. Amer. Math. Soc., Providence, R.I. 1963. MR 27, 6187.

    Google Scholar 

  108. Grünbaum, B. Borsuk’s problem and related questions. Proc. Sympos. Pure Math., Vol. VII pp. 271–284. Amer. Math. Soc., Providence, R.I. 1963. MR 27, 4134.

    Google Scholar 

  109. Grünbaum, B. Self-circumference of convex sets. Colloq. Math. 13 (1965), 55–57. MR 30, 2396.

    Google Scholar 

  110. Hammer, J. Problem 5368, Amer. Math. Monthly 73 (1966), 206.

    Google Scholar 

  111. Hammer, J. Unsolved problems concerning lattice points. Research Notes in Mathematics, vol. 15. Pitman, London-San Francisco-Melbourne, 1977. MR 56, 16515.

    Google Scholar 

  112. Hammer, P.C. Convex bodies associated with a convex body, Proc. Amer. Math. Soc. 2 (1951), 781–793. MR 14, p. 678.

    Google Scholar 

  113. Hammer, P.C. Diameters of convex bodies. Proc. Amer. Math. Soc. 5 (1954), 304–306. MR 15, p. 819.

    Google Scholar 

  114. Hammer, P.C. Constant breadth curves in the plane. Proc. Amer. Math. Soc. 6 (1955), 333–334. MR 16, p. 950.

    Google Scholar 

  115. Hammer, P.C. Convex curves of constant Minkowski breadth. Proc. Sympos. Pure Math., Vol. VII pp. 291–304. Amer. Math. Soc., Providence, R.I. 1963. MR 27, 4136.

    Google Scholar 

  116. Hammer, P.C., T.J. Smith Conditions equivalent to central symmetry of convex curves. Proc. Cambridge Philos. Soc. 60 (1964), 779–785. MR 30, 506.

    Google Scholar 

  117. Hammer, P.C., A. Sobczyk Planar line families I. Proc. Amer. Math. Soc. 4 (1953), 226–233. MR 14, p. 787.

    Google Scholar 

  118. Hammer, P.C., A. Sobczyk Planar line families II. Proc. Amer. Math. Soc. 4 (1953), 341–349. MR 15, p. 149. Hansen, H.C.

    Google Scholar 

  119. Hammer, P.C., A. Sobczyk: Et isoperimetrisk problem. Nordisk Math. Tidskr. 25/26 (1978), no. 3–4, 181–184, 210. MR 81a:52011.

    Google Scholar 

  120. Heil, E. Abschätzungen für einige Affininvarianten konvexer Kurven. Monatsh. Math. 71 (1967), 405–423. MR 37, 5796.

    Google Scholar 

  121. Heil, E. Verschärfungen des Vierscheitelsatzes und ihre relativ-geometrischen Verallgemeinerungen. Math. Nachr. 45 (1970), 227–241. MR 42, 5158.

    Google Scholar 

  122. Heil, E. Kleinste konvexe Körper gegebener Dicke. Preprint Nr. 453, Technische Hochschule Darmstadt, 1978.

    Google Scholar 

  123. Heil, E. Existenz eines 6-Normalenpunktes in einem konvexen Körper. Arch. Math. (Basel) 32 (1979), no. 4, 412–416. MR 80j:52002a.

    Google Scholar 

  124. Heil, E. Korrektur zu: “Existenz eines 6-Normalenpunktes in einem konvexen Körper”. Arch. Math. (Basel) 33 (1979/80), no. 5, 496. MR 80j:52002b.

    Google Scholar 

  125. Heppes, A. On characterization of curves of constant width. Mat. Lapok 10 (1959), 133–135 (Hungarian, Russian and English summaries). MR 22, 1846.

    Google Scholar 

  126. Herda, H. A characterization of circles and other closed curves. Amer. Math. Monthly 81(1974), 146–149. MR 48, 12290.

    Google Scholar 

  127. Hirakawa, J. On a characteristic property of the circle. Tôhoku Math. J. 37 (1933), 175–178. Zbl. 7, p. 318.

    Google Scholar 

  128. Hirakawa, J. On the relative breadth. Proc. Phys.-Math. Soc. Jap. III, 17 (1935), 137–144. Zbl. 11, p. 223.

    Google Scholar 

  129. Hirakawa, J. The affine breadth and its relation to the relative breadth. Japan. J. Math. 12 (1935), 43–50. Zbl. 12, p. 272.

    Google Scholar 

  130. a Hortobägyi, I. Über die Anzahl der Scheiben konstanter Breite, die eine gegebene Scheibe Konstanter Breite berühren. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 19 (1976), 45–48 (1977). MR 58, 24004.

    Google Scholar 

  131. Hortobägyi, I. Über die Durchlässigkeit einer aus Scheiben konstanter Breite bestehenden Schicht. Studia Sci. Math. Hungar. 11 (1976), no. 3–4, 383–387 (1978). MR 80c:52014.

    Google Scholar 

  132. Hoschek, J. Räumliche Zindlerkurven. Math. Balkanica 4 (1974), 253–260. MR 51, 9000.

    Google Scholar 

  133. Hsiao, E.K. Notes on differential geometry. Amer. Math. Monthly 66 (1959), 898–899. MR 21, 7540.

    Google Scholar 

  134. Irwin, M.C. Transnormal circles. J. London Math. Soc. 42 (1967), 545–552. MR 35, 4822.

    Google Scholar 

  135. Kallay, M. Reconstruction of a plane convex body from the curvature of its boundary. Israel J. Math. 17 (1974), 149–161. MR 50, 3110.

    Google Scholar 

  136. Kallay, M. The extrema] bodies in the set of plane bodies with a given width function. Israel J. Math. 22 (1975), 203–207. MR 53, 9033.

    Google Scholar 

  137. Kameneckii, I.M. Solution of a geometric problem of L. Ljusternik. (Russian). Uspehi Mat. Nauk 2 no. 2 (18), (1947), 199–202. MR 10, p. 60.

    Google Scholar 

  138. Kárteszi, F. Sur les figures convexes enveloppées par des carrées. Köz. Mat. Lapok 18 (1959), 1–6, 33–37 (Hungarian). MR 21, 5935.

    Google Scholar 

  139. Kárteszi, F. Über ein elementargeometrisches Problem. Ann. Univ. Sci. Budapest. Eötvös, Sect. Math. 2 (1959), 49–60. MR 22, 5933.

    Google Scholar 

  140. Kearsley, M.J. Curves of constant diameter. Math. Gaz. 36 (1952), 176–179. MR 14, p. 198.

    Google Scholar 

  141. Kelly, P.J. On Minkowski bodies of constant width. Bull. Amer. Math. Soc. 55 (1949), 1147–1150. MR 11, p. 386.

    Google Scholar 

  142. Kelly, P.J. Curves with a kind of constant width. Amer. Math. Monthly 64 (1957), 333–336. MR 19, p. 1073.

    Google Scholar 

  143. Khassa, D.S. Relation between maximal chords and symmetry for convex sets. J. London Math. Soc. 15 (1977), 541–546. MR 56, 3745.

    Google Scholar 

  144. Klee, V. Can a plane convex body have two equichordal points? Amer. Math. Monthly 76 (1969), 54–55.

    Google Scholar 

  145. Klee, V. Is a body spherical if its HA-measurements are constant? Amer. Math. Monthly 76 (1969), 539–542.

    Google Scholar 

  146. Klee, V. What is a convex set? Amer. Math. Monthly 78 (1971), 616–631. MR 44, 3202.

    Google Scholar 

  147. Klee, V. Some unsolved problems in plane geometry. Math. Mag. 52 (1979) no. 3, 131–145. MR 80m:52006.

    Google Scholar 

  148. Klötzler, R. Beweis einer Vermutung über n-Orbiformen kleinsten Inhalts (English and Russian summaries). Z. Angew. Math. Mech. 55 (1975), no. 10, 557–570. MR 53, 3884. Knuth, E.

    Google Scholar 

  149. Klötzler, R. On a problem of Klee, Acta Math. Acad. Sci. Hungar. 20 (1969), 169–177. MR 39, 861.

    Google Scholar 

  150. Krautwald, W. Kennzeichnungen der affinen Bilder von Körpern konstanter Breite. J. Geometry 15/2 (1980), 140–148. Zbl. 457, 52001.

    Google Scholar 

  151. Kubota, T. Einige Ungleichungen für die Eilinien und Eiflächen. Proc. Japan Acad. 24 no. 7–8 (1948), 1–3. MR 14, p. 678.

    Google Scholar 

  152. Kubota, T., D. Hemmi Some problems of minima concerning the oval. J. Math. Soc. Japan 5 (1953), 372–389. MR 15, p. 981.

    Google Scholar 

  153. Kuiper, N.H. Double normals of convex bodies. Israel J. Math. 2 (1964), 71–80. MR 30, 4191.

    Google Scholar 

  154. Kurbanov, N. Curves of constant width. (Russian. Tajiki summary). Izv. Akad. Nauk Turkmen. SSR Ser. Fiz.-Tehn. Him. Geol. Nauk 1975, no. 6, 3–8. MR 54, 3580.

    Google Scholar 

  155. Lay, S.R. Convex sets and their applications. Pure and applied mathematics, “A Wiley-Interscience publication”. J. Wiley and Sons, New York-Chichester-Brisbane-Toronto-Singapore 1982.

    Google Scholar 

  156. Lenz, H. Zur Zerlegung von Punktmengen in solche kleineren Durchmessers. Arch. Math. (Basel) 6 (1955), 413–416. MR 17, 887.

    Google Scholar 

  157. Lenz, H. Über die Bedeckung ebener Punktmengen durch solche kleineren Durchmessers. Arch. Math. (Basel) 7 (1956), 34–40. MR 17, p. 888.

    Google Scholar 

  158. Lewis, J.E. A Banach space whose elements are classes of sets of constant width. Canad. Math. Bull. 18 (1975), no. 5, 679–689. MR 58, 12289.

    Google Scholar 

  159. Loomis, P. Covering among constant relative width bodies in the plane. Ph.D. Dissertation, University of California, Davis, 1980.

    Google Scholar 

  160. Lutwak, E. On isoperimetric inequalities related to a problem of Moser. Amer. Math. Monthly 86 (1979), no. 6, 476–477. MR 80c:52010.

    Google Scholar 

  161. Isoperimetric inequalities involving bisectors, Bull. London Math. Soc. 12 (1980), 289–295. MR 81k:52024.

    Google Scholar 

  162. Lutwak, E. On packing curves into circles. Convexity and Related Combinatorial Geometry. Proceedings of the second University of Oklahoma Conference, pp. 107–111. Marcel Dekker, Inc., New York and Basel 1982.

    Google Scholar 

  163. Makuha, N.P. A connection between the inner and outer diameters of a general closed convex surface. (Russian). Ukrain. Geometr. Sb. Vyp. 2 (1966), 49–51. MR 35, 3550.

    Google Scholar 

  164. Mayer, A.E. Der Inhalt der Gleichdicke. Abschätzungen für ebene Gleichdicke. Math. Ann. 110 (1934), 97–127. Zbl. 9, p. 321.

    Google Scholar 

  165. Mayer, A.E. Über Gleichdicke kleinsten Flächeninhalts. Ani Akad. Wiss. Wien Nr. 7 (Sonderdruck), 4 pp. (1934). Zbl. 8, p. 404.

    Google Scholar 

  166. Mayer, A.E. Eine Uberkonvexität. Math. Z. 39 (1934), 512–531. Zbl. 10, p. 270.

    Google Scholar 

  167. Meissner, E. Über Punktmengen konstanter Breite. Vierteljahresschr. naturforsch. Ges Zürich 56 (1911), 42–50. Jbuch. 42, p. 91.

    Google Scholar 

  168. Meissner, E. Über die durch reguläre Polyeder nicht stützbaren Körper. Vierteljahresschr. naturforsch. Ges. Zürich 63 (1918), 544–551. Jbuch. 46, p. 1117.

    Google Scholar 

  169. Melzak, Z.A. A note on sets of constant width. Proc. Amer. Math. Soc. 11 (1960), 493–497. MR 23, A2124.

    Google Scholar 

  170. Melzak, Z.A. A property of plane sets of constant width. Canad. Math. Bull. 6 (1963), 409–415. MR 28, 3368.

    Google Scholar 

  171. Melzak, Z.A. Problems connected with convexity. Canad. Math. Bull. 8 (1965), 565–573. MR 33, 4781.

    Google Scholar 

  172. Melzak, Z.A. More problems connected with convexity. Canad. Math. Bull. 11 (1968), 489–494. MR 38, 3767.

    Google Scholar 

  173. Miller, K.G. Characterizations of the circle and the circular disk. Senior thesis, Macalester College, 1969.

    Google Scholar 

  174. Minkowski, H. Dichteste gitterförmige Lagerung kongruenter Körper. Nachrichten d. K. Ges. d. Wiss. Zu Göttingen, Math.-phys. Klasse 1904, 311–355 = Ges. Abh. Bd. 2, 1–42, Leipzig 1911. Minoda, T.

    Google Scholar 

  175. Minkowski, H. On the minimum of the perimeter of the circumrevolvable curve about an oval. Tôhoku Math. J. 45 (1939), 369–371. Zbl. 21, p. 67.

    Google Scholar 

  176. Minkowski, H. On the circle circumscribed about and the circle inscribed in an oval, and the sphere circumscribed about and the sphere inscribed in an ovaloid. Tôhoku Math. J. 45 (1939), 372–376. Zbl. 21, p. 67.

    Google Scholar 

  177. Minkowski, H. On certain ovals. Tôhoku Math. J. 48 (1941), 312–320. MR 10, p. 320.

    Google Scholar 

  178. Müller, H.R. Trochoidenhüllbahnen und Rotationskolbenmaschinen. Selecta Mathematica, III, pp. 119-

    Google Scholar 

  179. Müller, H.R. Heidelberger Taschenbucher, 86. Springer, Berlin 1971. MR 58, 30783.

    Google Scholar 

  180. Müller, H.R. Kurven konstanter Breite und Schiebkurven. Abh. Braunschweig. Wiss. Gesellsch. 26 (1976), 119–122. MR 55, 8965.

    Google Scholar 

  181. Nadeník, Z. Les courbes gauches de largeur constante. Czech. Math. J. 17 (92) (1967), 540–549. MR 36, 3242.

    Google Scholar 

  182. Ohmann, D. Extremalprobleme für konvexe Bereiche der euklidischen Ebene. Math. Z. 55 (1952), 346–352. MR 14, p. 76.

    Google Scholar 

  183. Petermann, E. Some relations between breadth and mixed volumes of convex bodies in R. Proc. Colloquium on Convexity (Copenhagen, 1965), pp. 229–233. K¢benhavns Univ. Inst. Copenhagen, 1967. MR 35, 6033.

    Google Scholar 

  184. Peterson, B.B. Do self-intersections characterize curves of constant width? Amer. Math. Monthly 79 (1972), 505–506. Zbl. 257, 52004.

    Google Scholar 

  185. Peterson, B.B. ntersection properties of curves of constant width. Illinois J. Math. 17 (1973), 411–420. MR 47, 9418.

    Google Scholar 

  186. Petty, C.M. On the geometry of the Minkowski plane. Riv. Math. Univ. Parma. 6 (1955), 269–292. MR 18, p. 760.

    Google Scholar 

  187. Petty, C.M. Isoperimetric problems. Proceedings of the Conference on Convexity and Combinatorial Geometry (Univ. of Oklahoma, Norman, Okla., 1971), (1972), 26–41. MR 50, 14499.

    Google Scholar 

  188. Petty, C.M. Geominimal surface area. Geom. Dedicata 3 (1974), 77–97. MR 48, 12313.

    Google Scholar 

  189. Petty, C.M., J.M. Crotty Characterization of spherical neighborhoods. Canad. J. Math. 22 (1970), 431–435. MR 41, 2538.

    Google Scholar 

  190. Petty, C.M., J.R. McKinney Convex bodies with circumscribing boxes of constant volume. To appear.

    Google Scholar 

  191. Porcu, L. Propriety metriche e affini di notevoli classi di ovali. Ist. Lombardo Accad. Sci. Lett. Rend. A 97 (1963), 899–929. MR 30, 2400.

    Google Scholar 

  192. Rademacher, H., O. Toeplitz The Enjoyment of Mathematics. Princeton Univ. Press, Princeton, N.J. 1957 (Translation of “Von Zahlen und Figuren”, Springer, Berlin 1933). MR 18, p. 454.

    Google Scholar 

  193. Reidemeister, K. Über Körper konstanten Durchmessers. Math. Z. 10 (1921), 214–216. Jbuch. 48, p. 835.

    Google Scholar 

  194. Reinhardt, K. Extremale Polygone gegebenen Durchmessers. Jahresber. Deutsch. Math.-Verein. 31(1922), 251–270.

    Google Scholar 

  195. Robertson, S.A. Generalized constant width for manifolds. Michigan Math. J. 11 (1964), 97–105. MR 29, 2815.

    Google Scholar 

  196. Robertson, S.A. On transnormal manifolds. Topology 6 (1967), 117–123. MR 34, 6793.

    Google Scholar 

  197. Rogers, C.A. Symmetrical sets of constant width and their partitions. Mathematika 18 (1971), 105–111. MR 44, 7432.

    Google Scholar 

  198. Sachs, H. Über eine Klasse isoperimetrischer Probleme. I, II, III. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe 8 (1958/59), 121–126, 127–134, 345–350. MR 21, 7482/7483.

    Google Scholar 

  199. Sallee, G.T. The maximal set of constant width in a lattice. Pacific J. Math. 28 (1969), 669–674. MR 39, 2069.

    Google Scholar 

  200. Sallee, G.T. Maximal areas of Reuleaux polygons. Canad. Math. Bull. 13 (1970), 175–179. MR 42, 2368.

    Google Scholar 

  201. Sallee, G.T. Reuleaux polytopes. Mathematika 17 (1970), 315–328. MR 45, 5872.

    Google Scholar 

  202. Sallee, G.T. An inequality for curves. Unpublished manuscript.

    Google Scholar 

  203. Sancho de San Roman, J.: On closed skew curves, especially of constant width. (Spanish). Memorias de Matematica del Instituto “Jorge Juan” no. 10 (1949), 67 pp. MR 11, p. 201.

    Google Scholar 

  204. Sancho de San Roman, J. Curvas alabeadas de anchura afin constante. Collect. Math. 8 (1955–1956), 85–98. MR 18, p. 760.

    Google Scholar 

  205. Sancho de San Roman, J.Sobre un nuevo concepto de anchura de óvalos, invariante afin. Mat. Hisp: Amer. (4) 16 (1956), 151–171. MR 18, p. 505.

    Google Scholar 

  206. Sancho de San Roman, J.Un nuevo concepto de anchura afin de cuerpos ovales. Acad. Ci. Madrid 51 (1957), 229–244. MR 20, 2666.

    Google Scholar 

  207. Sancho de San Roman, J.Poligonos de anchura afin constante. Fourth Annual Meeting Spanish Mathematicians, pp. 107–108. Univ. Salamanca, Salamanca 1965. MR 33, 1795. Santaló, L.A.

    Google Scholar 

  208. Sancho de San Roman, J.Propriedades de las figuras convexas sobre la esfera. Math. Notae 4 (1944), 11–40. MR 6, p. 15.

    Google Scholar 

  209. b Sancho de San Roman, J.Note on convex spherical curves. Bull. Amer. Math. Soc. 50 (1944), 528–534. MR 6, p. 100.

    Google Scholar 

  210. Sancho de San Roman, J.Note on convex curves on the hyperbolic plane. Bull. Amer. Math. Soc. 51(1945), 405–412. MR 7, p. 26.

    Google Scholar 

  211. Sancho de San Roman, J.Sobre los cuerpos convexos de anchura constante en E“. Portugaliae Math. 5 (1946), 195–201. MR 9, p. 526.

    Google Scholar 

  212. Sancho de San Roman, J.On parallel hypersurfaces in the elliptic and hyperbolic n-dimensional space. Proc. Amer. Math. Soc. 1 (1950), 325–330. MR 12, p. 124.

    Google Scholar 

  213. Sancho de San Roman, J.Sobre systemas completos de desigualidades entre tres elementos de una figura convexa plana. Math. Notae 17 (1959/61), 82–104. MR 26, 6857.

    Google Scholar 

  214. Sancho de San Roman, J.Integral geometry and geometric probability. Encyclopedia of Mathematics and its Applications, Vol. 1. Addison-Wesley Publ. Co., Reading, Mass.-London-Amsterdam, 1976. MR 55, 6340.

    Google Scholar 

  215. Schaal, H. Prüfung einer Kreisform mit Hohlwinkel und Taster. Elem. Math. 17 (1962), 33–38. MR 25, 1489.

    Google Scholar 

  216. Schaal, H. Gleitkurven in regulären Polygonen. Zeitschr. Angew. Math. Mech. 43 (1963), 459–476. (English and Russian summaries). MR 28, 1800.

    Google Scholar 

  217. Schaer, J., J.E. Wetzel Boxes for curves of constant length. Israel J. Math. 12 (1972), 257–265. MR 47, 5726.

    Google Scholar 

  218. Schneider, R. Gleitkörper in konvexen Polytopen. J. Reine Angew. Math. 248 (1971), 193–220. MR 43, 5411.

    Google Scholar 

  219. Schneider, R. Boundary structure and curvature of convex bodies. Contributions to Geometry (Proc. Geom. Sympos., Siegen 1978), pp. 13–59, Birkhäuser, Basei 1979. MR 81i:52001. Schopp, J.

    Google Scholar 

  220. Schneider, R. Über die Newtonsche Zahl einer Scheibe konstanter Breite. Studia Sci. Math. Hungar. 5 (1970), 475–478. MR 44, 3200.

    Google Scholar 

  221. Schneider, R. Üeberdeckungsprobleme ebener Bereiche von konstantem Durchmesser. Period. Polytechn. Mech. Engrg. 21 (1977), no. 2, 103–109. MR 58, 7408.

    Google Scholar 

  222. Schulte, E. Konstruktion regulärer Hüllen konstanter Breite. Monatsh. Math. 92 (1981), 313–322.

    Google Scholar 

  223. Scott, P.R. Two inequalities for convex sets in the plane. Bull. Austral. Math. Soc. 19 (1978), no. 1, 131–133. MR 80d:52005.

    Google Scholar 

  224. Scott, P.R. Sets of constant width and inequalities. Quart. J. Math. Oxford (2) 32 (1981), 345–348.

    Google Scholar 

  225. Sholander, M. On certain minimum problems in the theory of convex curves. Trans. Amer. Math. Soc. 73 (1952), 139–173. MR 14, p. 787.

    Google Scholar 

  226. Silverman, R. Decomposition of plane convex sets II. Sets associated with a width function. Pacific J. Math. 48 (1973), 497–502. MR 54, 3581.

    Google Scholar 

  227. Sine, R. A characterization of the ball in R3. Amer. Math. Monthly 83 (1976), 260–261. MR 53, 1404.

    Google Scholar 

  228. Sine R., V. Kreinovic Remarks on billiards. Amer. Math. Monthly 86 (1979), 204–206. MR 80k:28021.

    Google Scholar 

  229. Smakal, S. Curves of constant width. Czech. Math. J. 23 (98) (1973), 86–94. MR 47, 4154.

    Google Scholar 

  230. Smith, T.J. Planar line families. Ph.D. Thesis, University of Wisconsin, 1961.

    Google Scholar 

  231. Soltan, V.P. A theorem on full sets (Russian). Dokl. Akad. Nauk. SSSR 234 (1977), no. 2, 320–322. MR 58, 2607.

    Google Scholar 

  232. Soltan, V.P. Bodies of constant width in finite-dimensional normed spaces (Russian). Mat. Zametki 25 (1979), no. 2, 283–291, 319. MR 80d:52004.

    Google Scholar 

  233. Su, B. On the curvature-axis of the convex closed curve. Sci. Rep. Tôhoku Univ. 17 (1928), 35–42. Jbuch. 54, p. 799.

    Google Scholar 

  234. Süss, W. Über affine Geometrie XL: Eiflächen konstanter Affinbreite. Math. Ann. 96 (1927) 251–260. Jbuch. 53, p. 713.

    Google Scholar 

  235. Sz.-Nagy, G. Zentralsymmetrisierung konvexer Körper. Publ. Math. Debrecen 1 (1949), 29–32. MR 11, p. 386.

    Google Scholar 

  236. Tanno, S. Cm-approximation of continuous ovals of constant width. J. Math. Soc. Japan 28 (1976), no. 2, 384–395. MR 53, 1489.

    Google Scholar 

  237. Taylor, S.J. Some simple geometrical extremal problems. Math. Gaz. 37 (1953), 188–198. MR 15, p. 247.

    Google Scholar 

  238. Tennison, R.L. Smooth curves of constant width. Math. Gaz. 60 (1976), 270–272. Zbl. 341, 53002.

    Google Scholar 

  239. Thüring, R. Studien über den Holditchschen Satz. Verh. Naturforsch. Ges. Basel 63 (1952) 221–252. MR 14, p. 679.

    Google Scholar 

  240. Togliatti, E. Sulle curve piane di larghezza costante. Period. Mat. (4) 46 (1968), 359–371. MR 38, 6457.

    Google Scholar 

  241. Toranzos, F.A. Sobre un problema de B. Grünbaum. Rev. Un. Mat. Argentina 22 (1964), 103–106. MR 30, 507.

    Google Scholar 

  242. Toranzos, F.A., J.H. Nanclares Convexidad. Cursos, Seminarios y Tesis del PEAM, No. 4, PEAM, Universidad del Zulia, Maracaibo, 1978, 155 pp. MR 58, 24001.

    Google Scholar 

  243. Valette, G. Couples de corps convexes dont la difference des largeurs est constante. Collection of articles honoring P. Burniat, Th. Lepage and P. Libois. Bull. Soc. Math. Belg. 23 (1971), 493–499. MR 48, 1037.

    Google Scholar 

  244. Varela Gil, J. Sobre generación de curvas orbiformes. Atti del Congresso Internazionale dei Matematici, Bologna 1928. Vol. V, pp. 483–488, Bologna 1932.

    Google Scholar 

  245. Varela Gil, J. Curvas orbiformes multiples. Univ. Nat. Tucumán. Revista A. 4 (1944). MR 7, p. 168.

    Google Scholar 

  246. Vincensini, P. Sur les corps convexes admettant un domaine vectoriel donné. C.R. Acad. Sci. Paris 201 (1935), 761–763. Zbl. 12, p. 272.

    Google Scholar 

  247. Vincensini, P. Sur le prolongement des séries linéaires de corp convexes. Applications. Rend. Circ. mat. Palermo 60 (1936), 361–372. Zbl. 17, p. 230.

    Google Scholar 

  248. Vincensini, P. Sur les corps de largeur constante de l’espace à trois dimensions. C.R. Acad. Sci. Paris 204 (1937), 84–86. Zbl. 15, p. 368.

    Google Scholar 

  249. Vincensini, P. Corps convexes. Series linéaires. Domaines vectoriels. Mem. des Sciences Math., fasc. XCIV, Paris 1938. Zbl. 21, p. 356.

    Google Scholar 

  250. Vincensini, P. Sur l’application d’une méthode géométrique à l’étude de certains ensembles de corps convexes. Colloque sur les questions de réalité en géométrie, Liège 1955, pp. 77–79, Paris 1956. MR 17, p. 1123.

    Google Scholar 

  251. Vincensini, P. Sur un invariant de partition de l’ensemble des corps convexes de l’espace euclidien a n-dimension. C.R. Acad. Sci. Paris 245 (1957), 132–133. MR 19, p. 573.

    Google Scholar 

  252. Vivanti, G. Sulle curve piane a normali doppie. Ist. Lombardo Sci. Lett. Rend. Cl. Sci. Mat. (3) 10 (79) (1946), 256–260. MR 10, p. 325.

    Google Scholar 

  253. Vrecica, S. A note on sets of constant width. Publications de L’institut Mathematique, Nouvelle série 29 (1981), 289–291.

    Google Scholar 

  254. Wajnstejn, D. Courbes à largeur constante. (Polish). Wiadom. mat. 45 (1938), 39–63. Zbl. 19, p. 140.

    Google Scholar 

  255. Watson, A.G.D. On Mizel’s problem. J. London Math. Soc. 37 (1962). 307–308. MR 27, 4141a.

    Google Scholar 

  256. Wegner, B. Globale Sätze über Raumkurven konstanter Breite. Math. Nachr. 53 (1972), 337–344. MR 47, 5735.

    Google Scholar 

  257. Wegner, B. Globale Sätze über Raumkurven konstanter Breite II. Math. Nachr. 67 (1975), 213–223. MR 51, 11380.

    Google Scholar 

  258. Wegner, B. Verallgemeinerte Zindlerkurven in euklidischen Räumen. Math. Balkanica 6 (1976), 298–306 (1978). MR 80m:53002.

    Google Scholar 

  259. Wegner, B. Analytic approximation of continuous ovals of constant width. J. Math. Soc. Japan 29 (1977), no. 3, 537–540. MR 57, 4013.

    Google Scholar 

  260. Weissbach, B. Rotoren im regulären Dreieck. Publ. Math. Debrecen 19 (1972), 21–27 (1972). MR 49, 3685.

    Google Scholar 

  261. Weissbach, B. Zur Inhaltsschätzung von Eibereichen. Wiss. Beitr. Martin-Luther-Univ. Halle-Wittenberg M 8 Beitr. Algebra Geom. 6 (1977), 27–35. MR 58, 7410.

    Google Scholar 

  262. Yaglom, I.M., V.G. Boltyanskii Convex figures. Holt, Rinehart and Winston, New York, 1961. MR 23, Al283.

    Google Scholar 

  263. Zindler, K. Über konvexe Gebilde, II. Teil. Monatsh. Math. Phys. 31 (1921), 25–56. Jbuch. 48, p. 833.

    Google Scholar 

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  1. Department of Mathematics, University of California Davis, Davis, CA, 95616, USA

    G. D. Chakerian

  2. Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    H. Groemer

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    Jörg M. Wills

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Chakerian, G.D., Groemer, H. (1983). Convex Bodies of Constant Width. In: Gruber, P.M., Wills, J.M. (eds) Convexity and Its Applications. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5858-8_3

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