Advertisement

Characteristic properties of ellipsoids and convex quadrics

  • Valeriu Soltan
Article

Abstract

This is a survey on various characteristic properties of ellipsoids and convex quadrics in the family of convex hypersurfaces in \({\mathbb {R}}^n\). The topics under consideration include planar sections and projections, planarity conditions on midsurfaces and shadow-boundaries, intersections of homothetic copies, projective centers, and invariant mappings.

Keywords

Convex Curve Body Solid Homothetic Hypersurface Invariant Midsurface Projection Quadric Section Shadow-boundary Surface Symmetric 

Mathematics Subject Classification

Primary 52A20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author thanks the anonymous referees for many helpful suggestions which improved the original version of the manuscript.

References

  1. 1.
    Aitchison, P.W.: A characterization of the ellipsoid. J. Aust. Math. Soc. 11, 385–394 (1970)zbMATHCrossRefGoogle Scholar
  2. 2.
    Aitchison, P.W.: The determination of convex bodies by some local conditions. Duke Math. J. 41, 193–209 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Aitchison, P.W., Petty, C.M., Rogers, C.A.: A convex body with a false centre is an ellipsoid. Mathematika 18, 50–59 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Alexandrov, A.D.: On convex surfaces with plane shadow-boundaries. Mat. Sbornik 5, 309–316 (1939)Google Scholar
  5. 5.
    Alonso, J., Martín, P.: Some characterizations of ellipsoids by sections. Discrete Comput. Geom. 31, 643–654 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Alonso, J., Martín, P.: Convex bodies with sheafs of elliptic sections. J. Convex Anal. 13, 169–175 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Alonso, J., Martín, P.: Convex bodies with sheafs of elliptic sections. II. J. Convex Anal. 14, 1–11 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Amir, D.: Characterizations of Inner Product Spaces. Birkhäuser, Basel (1986)zbMATHCrossRefGoogle Scholar
  9. 9.
    Arelio, I., Montejano, L.: Convex bodies with many elliptic sections. J. Convex Anal. 24, 685–693 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Arocha, J.L., Montejano, L., Morales, E.: A quick proof of Höbinger–Burton–Larman’s theorem. Geom. Dedicata 63, 331–335 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Auerbach, H.: Sur les groupes bornés de substitutions linéaires. C. R. Acad. Sci Paris 195, 1367–1369 (1932)zbMATHGoogle Scholar
  12. 12.
    Auerbach, H.: Sur une propriété carastéristique de l’ellipsoïde. Studia Math. 9, 17–22 (1940)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Auerbach, H., Mazur, S., Ulam, S.: Sur une propriété caractéristique de l’ellipsoïde. Monatsh. Math. 42, 45–48 (1935)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Berger, K.H.: Eilinien mit perspektiv liegenden Tangenten-und Sehnendreiecken. S.-B. Heidelberg. Akad. Wiss. 1–11 (1936)Google Scholar
  15. 15.
    Berger, M.: Geometry. I, II. Springer, Berlin (1987)zbMATHCrossRefGoogle Scholar
  16. 16.
    Bertrand, J.: Démonstration d’un théoreme de géométrie. J. Math. Pures Appl. 7, 215–216 (1842)Google Scholar
  17. 17.
    Besicovitch, A.S.: A problem on a circle. J. Lond. Math. Soc. 36, 241–244 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Bianchi, G., Gruber, P.M.: Characterization of ellipsoids. Arch. Math. (Basel) 49, 344–350 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Blaschke, W.: Räumliche Variationsprobleme mit symmetrischen Transversalitätsbedingungen. Ber. Math. Phys. Kl. Königl. Sächs. Ges. Wiss. Leipzig 68, 50–55 (1916)Google Scholar
  20. 20.
    Blaschke, W.: Kreis und Kugel. Veit, Leipzig (1916)zbMATHGoogle Scholar
  21. 21.
    Blaschke, W.: Altes und Neues von Ellipse und Ellipsoid. Jahresber. Deutsch. Math. Vereinig. 26, 220–230 (1917)zbMATHGoogle Scholar
  22. 22.
    Blaschke, W.: Vorlesungen über Differentialgeometrie II. Affine Differentialgeometrie. Springer, Berlin (1923)zbMATHGoogle Scholar
  23. 23.
    Blaschke, W.: Zur Affingeometrie der Eilinien und Elfächen. Math. Nachr. 15, 258–264 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Blaschke, W., Hessenberg, G.: Lehrsätze über konvexe Körper. Jahresber. Deutsch. Math.-Vereinig. 26, 215–220 (1917)zbMATHGoogle Scholar
  25. 25.
    Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper. Springer, Berlin, 1934. English translation: Theory of convex bodies. BCS Associates, Moscow, ID (1987)Google Scholar
  26. 26.
    Borodin, P.A.: Quasi-orthogonal sets and conditions for the Hilbert property of a Banach space. Sb. Math. 188, 1171–1182 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Borodin, P.A.: A new proof of Blaschke’s ellipsoid theorem. Mosc. Univ. Math. Bull. 58(3), 6–10 (2003)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Brunn, H.: Über Kurven ohne Wendepunkte. Habilitationschrift, Ackermann, München (1889)Google Scholar
  29. 29.
    Burton, G.R.: Sections of convex bodies. J. Lond. Math. Soc. 12, 331–336 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Burton, G.R.: On the sum of a zonotope and an ellipsoid. Comment. Math. Helv. 51, 369–387 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Burton, G.R.: Some characterisations of the ellipsoid. Israel J. Math. 28, 339–349 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Burton, G.R.: Congruent sections of a convex body. Pac. J. Math. 81, 303–316 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Burton, G.R., Larman, D.G.: On a problem of J. Höbinger. Geom. Dedicata 5, 31–42 (1976)zbMATHCrossRefGoogle Scholar
  34. 34.
    Burton, G.R., Mani, P.: A characterization of the ellipsoid in terms of concurent sections. Comment. Math. Helv. 53, 485–507 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Busemann, H.: The Geometry of Geodesics. Academic Press, New York (1955)zbMATHGoogle Scholar
  36. 36.
    Busemann, H.: Timelike spaces. Dissertationes Math. (Rozprawy Mat.) 53, p. 52 (1967)Google Scholar
  37. 37.
    Carathéodory, C.: Über den Variabilitätsbereich der Koefficienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann. 64, 95–115 (1907)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Chakerian, G.D.: The affine image of a convex body of constant breadth. Israel J. Math. 3, 19–22 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Coolidge, J.L.: A History of the Conic Sections and Quadric Surfaces. Oxford University Press, Oxford (1945)zbMATHGoogle Scholar
  40. 40.
    Danzer, L.W.: A characterization of the circle. In: Klee, V. (ed.) Convexity, pp. 99–100. American Mathematical Society, Providence (1963)CrossRefGoogle Scholar
  41. 41.
    Düvelmeyer, N.: Convex bodies with equiframed two-dimensional sections. Arch. Math. (Basel) 88, 181–192 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Gardner, R.J.: Geometric Tomography. Cambridge University Press, New York (1995)zbMATHGoogle Scholar
  43. 43.
    Goodey, P.R.: Homothetic ellipsoids. Math. Proc. Camb. Philos. Soc. 93, 25–34 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Gromov, M.L.: On a geometric hypothesis of Banach. Izv. Akad. Nauk. SSSR Ser. Mat. 31, 1105–1114 (1967)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Gruber, P.M.: Über kennzeichende Eigenschaften von eucklidischen Räumen und Ellipsoiden. I. J. Reine Angew. Math. 265, 61–83 (1974)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Gruber, P.M.: Über kennzeichnende Eigenschaften von eukclidischen Räumen und Ellipsoiden. II. J. Reine Angew. Math. 270, 123–142 (1974)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Gruber, P.M.: Über kennzeichende Eigenschaften von eucklidischen Räumen und Ellipsoiden. III. Monatsh. Math. 78, 311–340 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Gruber, P.M.: A Helmholtz-Lie type characterization of ellipsoids. I. Discrete Comput. Geom. 13, 517–527 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Gruber, P.M., Höbinger, J.: Kennzeichnungen von Ellipsoiden mit Anwendungen. In: U. Kulisch, et al. (eds), Jahrbuch Überblicke Mathematik, pp. 9–29, 1976. Bibliographisches Inst. Mannheim (1976)Google Scholar
  50. 50.
    Gruber, P.M., Ludwig, M.: A Helmholtz–Lie type characterization of ellipsoids. II. Discrete Comput. Geom. 16, 55–67 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Gruber, P.M., Ódor, T.: Ellipsoids are the most symmetric convex bodies. Arch. Math. (Basel) 73, 394–400 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Grünbaum, B.: Arrangements and Spreads. American Mathematical Society, Providence (1972)zbMATHCrossRefGoogle Scholar
  53. 53.
    Hadwiger, H.: Vollständig stetige Umwendung ebener Eibereiche im Raum. In: Szegö, G. (ed.) Studies in Mathematical Analysis and Related Topics, pp. 128–131. Stanford University Press, Stanford (1962)Google Scholar
  54. 54.
    Heil, E., Martini, H.: Special convex bodies. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, pp. 347–385. North-Holland, Amsterdam (1993)zbMATHCrossRefGoogle Scholar
  55. 55.
    Helmholtz, H.: Über die Tatsachen, die der Geometrie zu Grunde liegen, pp. 193–221. Göttingen Nachrichten, Göttingen (1868)zbMATHGoogle Scholar
  56. 56.
    Höbinger, J.: Über einen Satz von Aitchison, Petty und Rogers. Ph.D. Thesis. Techn. Univ. Wien (1974)Google Scholar
  57. 57.
    Ivanov, B.A.: Straight line segments on the boundary of a convex body. Ukrain. Geom. Sb. No. 13, 69–71 (1973)MathSciNetGoogle Scholar
  58. 58.
    Ivanov, S.: Monochromatic Finsler surfaces and a local ellipsoid characterization. Proc. Am. Math. Soc. 146, 1741–1755 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Jerónimo-Castro, J., McAllister, T.B.: Two characterizations of ellipsoidal cones. J. Convex Anal. 20, 1181–1187 (2013)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Kakeya, S.: On some properties of convex curves and surfaces. Tôhoku Math. J. 8, 218–221 (1915)zbMATHGoogle Scholar
  61. 61.
    Kakutani, S.: Some characterizations of Euclidean space. Jpn. J. Math. 15, 93–97 (1939)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Kelly, P., Straus, E.G.: On the projective centres of convex curves. Can. J. Math. 12, 568–581 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Klee, V.L.: Some characterizations of convex polyhedra. Acta Math. 102, 79–107 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Kneser, M.: Eibereiche mit geraden Schwerlinien. Math. Phys. Semesterber. 1, 97–98 (1949)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Kojima, T.: On characteristic properties of the conic and quadric. Sci. Rep. Tôhoku Univ. 8, 67–78 (1919)zbMATHGoogle Scholar
  66. 66.
    Kubota, T.: On the theory of closed convex surface. Proc. Lond. Math. Soc. 14, 230–239 (1914)zbMATHGoogle Scholar
  67. 67.
    Kubota, T.: Einfache Beweise eines Satzes über die konvexe geschlossene Fläche. Sci. Rep. Tôhoku Univ. 3, 235–255 (1914)zbMATHGoogle Scholar
  68. 68.
    Kubota, T.: Über die konvexe geschlossene Fläche. Sci. Rep. Tôhoku Univ. 3, 277–287 (1914)zbMATHGoogle Scholar
  69. 69.
    Kubota, T.: On a characteristic property of the ellipse. Tôhoku Math. J. 9, 148–151 (1916)zbMATHGoogle Scholar
  70. 70.
    Larman, D.G.: A note on the false centre problem. Mathematika 21, 216–227 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Larman, D., Montejano, L., Morales-Amaya, E.: Characterization of ellipsoids by means of parallel translated sections. Mathematika 56, 363–378 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Lenz, H.: Einige Anwendungen der projektiven Geometrie auf Fragen der Flächentheorie. Math. Nachr. 18, 346–359 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Lie, S.: Theorie der Transformationsgruppen. Bd. III. Teubner, Leipzig (1893)Google Scholar
  74. 74.
    Lie, S.: Bestimmung aller Flächen, die eine continuirliche Schar von projectiven Transformationen gestatten. Ber. Sächs. Akad. Wiss. Leipzig 47, 209–260 (1895)zbMATHGoogle Scholar
  75. 75.
    Makai, E., Soltan, V.: Lower bounds on the numbers of shadow-boundaries and illuminated regions of a convex body. In: Böröczky, K., et al. (eds.) Intuitive Geometry (Szeged, 1991), pp. 249–268. Colloq. Math. Soc. János Bolyai 63, North-Holland, Amsterdam, (1994)Google Scholar
  76. 76.
    Mani, P.: Fields of planar bodies tangent to spheres. Monatsh. Math. 74, 145–149 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    Marchaud, A.: Sur les ovales. Ann. Soc. Polon. 21, 324–331 (1948)MathSciNetzbMATHGoogle Scholar
  78. 78.
    Marchaud, A.: Un théorème sur les corps convexes. Ann. Scient. École Norm. Supér. 76, 283–304 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Martini, H., Soltan, V.: Combiatorial problems on the illumination of convex bodies. Aequ. Math. 57, 121–152 (1999)zbMATHCrossRefGoogle Scholar
  80. 80.
    Mauldin, R.D. (ed.): The Scottish Book. Birkäuser, Boston (1981)zbMATHGoogle Scholar
  81. 81.
    Mazur, S.: Quelques propriétés caractéristiques des espaces euclidiens. C. R. Acad. Sci. Paris 207, 761–764 (1938)zbMATHGoogle Scholar
  82. 82.
    Montejano, L.: Convex bodies with homothetic sections. Bull. Lond. Math. Soc. 23, 381–386 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Montejano, L., Morales-Amaya, E.: Characterization of ellipsoids and polarity in convex sets. Mathematika 50, 63–72 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Montejano, L., Morales-Amaya, E.: Variations of classic characterizations of ellipsoids and a short proof of the false centre theorem. Mathematika 54, 35–40 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Montejano, L., Morales-Amaya, E.: Shaken false centre theorem. I. Mathematika 54, 41–46 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Nakagawa, S.: On some theorems regarding ellipsoids. Tôhoku Math. J. 8, 11–13 (1915)zbMATHGoogle Scholar
  87. 87.
    Nakajima, S.: Eilinien mit geraden Schwerlinien. Jpn. J. Math. 5, 81–84 (1928)zbMATHCrossRefGoogle Scholar
  88. 88.
    Nakajima, S.: Über konvexe Kurven und Flächen. Tôhoku Math. J. 29, 227–230 (1928)zbMATHGoogle Scholar
  89. 89.
    Olovjanishnikov, S.P.: On a characterization of the ellipsoid. Učen. Zap. Leningrad. State Univ. Ser. Mat. 83, 114–128 (1941)Google Scholar
  90. 90.
    Petty, C.M.: Ellipsoids. In: Gruber, P.M., Wills, J.M. (eds.) Convexity and its Applications, pp. 264–276. Birkhäuser, Basel (1983)CrossRefGoogle Scholar
  91. 91.
    Phillips, R.S.: A characterization of Euclidean spaces. Bull. Am. Math. Soc. 46, 930–933 (1940)MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Rogers, C.A.: Sections and projections of convex bodies. Port. Math. 24, 99–103 (1965)MathSciNetzbMATHGoogle Scholar
  93. 93.
    Rudin, W., Smith, K.T.: Linearity of best approximation: a characterization of ellipsoids. Indag. Math. 23, 97–103 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  94. 94.
    Šaĭdenko, A.V.: Some characteristic properties of an ellipsoid. Sibirsk. Mat. Ž. 21, 232–234 (1980)MathSciNetGoogle Scholar
  95. 95.
    Socié-Méthou, E.: Caractérisation des ellipsoïdes par leurs groupes d’automorphismes. Ann. Sci. École Norm. Sup. 35, 537–548 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Soltan, V.: Convex bodies with polyhedral midhypersurfaces. Arch. Math. 65, 336–341 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Soltan, V.: Affine diameters of convex-bodies—a survey. Expo. Math. 23, 47–63 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    Soltan, V.: Convex solids with planar midsurfaces. Proc. Am. Math. Soc. 136, 1071–1081 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    Soltan, V.: Convex solids with homothetic sections through given points. J. Convex Anal. 16, 473–486 (2009)MathSciNetzbMATHGoogle Scholar
  100. 100.
    Soltan, V.: Convex quadrics. Bul. Acad. Ştiinţe Repub. Moldova. Mat. 3, 94–106 (2010)MathSciNetzbMATHGoogle Scholar
  101. 101.
    Soltan, V.: Convex solids with hyperplanar midsurfaces for restricted families of chords. Bul. Acad. Ştiinţe Repub. Moldova. Mat. 2, 23–40 (2011)MathSciNetzbMATHGoogle Scholar
  102. 102.
    Soltan, V.: Convex solids with hyperplanar shadow-boundaries. J. Convex Anal. 19, 591–607 (2012)MathSciNetzbMATHGoogle Scholar
  103. 103.
    Soltan, V.: Convex solids whose point-source shadow-boundaries lie in hyperplanes. J. Geom. 103, 149–160 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    Soltan, V.: Convex quadrics and their characterizations by means of plane sections. In: Toni, B. (ed.) Bridging Mathematics, Statistics, Engineering and Technology, pp. 131–145. Springer, Berlin (2012)CrossRefGoogle Scholar
  105. 105.
    Soltan, V.: Characterizations of convex quadrics in terms of plane quadric sections, midsurfaces, and shadow-boundaries. In: Toni, B. (ed.) New Frontiers of Multidisciplinary Research in STEAM-H, pp. 79–110. Springer, Berlin (2014)Google Scholar
  106. 106.
    Soltan, V.: Convex hypersurfaces with hyperplanar intersections of their homothetic copies. J. Convex Anal. 22, 145–159 (2015)MathSciNetzbMATHGoogle Scholar
  107. 107.
    Soltan, V.: Lectures on Convex Sets. World Scientific, Hackensack (2015)zbMATHCrossRefGoogle Scholar
  108. 108.
    Soltan, V.: Convex surfaces with planar polar sets and point-source shadow-boundaries. J. Convex Anal. 24, 645–660 (2017)MathSciNetzbMATHGoogle Scholar
  109. 109.
    Süss, W.: Kennzeichnende Eigenschaften der Kugel als Folgerung eines Browerschen Fixpunktsatzes. Comment. Math. Helv. 20, 61–64 (1947)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    Süss, W.: Eine elementare kennzeichnende Eigenschaft des Ellipsoids. Math.-Phys. Semesterber. 3, 57–58 (1953)MathSciNetzbMATHGoogle Scholar
  111. 111.
    Süss, W., Viet, U., Berger, K.H.: Konvexe Figuren. In: Behnke, H. (ed.) Grundzüge der Mathematik. Bd. II: Geometrie. Teil B, pp. 361–381. Vanderhoeck and Ruprecht, Göttingen (1960)Google Scholar
  112. 112.
    Tietze, H.: Über Konvexheit im kleinen und im grossen und über gewisse den Punkten einer Menge zugeordnete Dimensionszahlen. Math. Z. 28, 697–707 (1928)MathSciNetzbMATHCrossRefGoogle Scholar
  113. 113.
    Watson, A.G.D.: On Mizel’s problem. J. Lond. Math. Soc. 37, 307–308 (1962)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesGeorge Mason UniversityFairfaxUSA

Personalised recommendations