Skip to main content
SpringerLink
Log in

Characteristic properties of ellipsoids and convex quadrics

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aitchison, P.W.: A characterization of the ellipsoid. J. Aust. Math. Soc. 11, 385–394 (1970)

    MATH  Google Scholar 

  2. Aitchison, P.W.: The determination of convex bodies by some local conditions. Duke Math. J. 41, 193–209 (1974)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  4. Alexandrov, A.D.: On convex surfaces with plane shadow-boundaries. Mat. Sbornik 5, 309–316 (1939)

    Google Scholar 

  5. Alonso, J., Martín, P.: Some characterizations of ellipsoids by sections. Discrete Comput. Geom. 31, 643–654 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Alonso, J., Martín, P.: Convex bodies with sheafs of elliptic sections. J. Convex Anal. 13, 169–175 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Alonso, J., Martín, P.: Convex bodies with sheafs of elliptic sections. II. J. Convex Anal. 14, 1–11 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Amir, D.: Characterizations of Inner Product Spaces. Birkhäuser, Basel (1986)

    MATH  Google Scholar 

  9. Arelio, I., Montejano, L.: Convex bodies with many elliptic sections. J. Convex Anal. 24, 685–693 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Arocha, J.L., Montejano, L., Morales, E.: A quick proof of Höbinger–Burton–Larman’s theorem. Geom. Dedicata 63, 331–335 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Auerbach, H.: Sur les groupes bornés de substitutions linéaires. C. R. Acad. Sci Paris 195, 1367–1369 (1932)

    MATH  Google Scholar 

  12. Auerbach, H.: Sur une propriété carastéristique de l’ellipsoïde. Studia Math. 9, 17–22 (1940)

    MathSciNet  MATH  Google Scholar 

  13. Auerbach, H., Mazur, S., Ulam, S.: Sur une propriété caractéristique de l’ellipsoïde. Monatsh. Math. 42, 45–48 (1935)

    MathSciNet  MATH  Google Scholar 

  14. Berger, K.H.: Eilinien mit perspektiv liegenden Tangenten-und Sehnendreiecken. S.-B. Heidelberg. Akad. Wiss. 1–11 (1936)

  15. Berger, M.: Geometry. I, II. Springer, Berlin (1987)

    MATH  Google Scholar 

  16. Bertrand, J.: Démonstration d’un théoreme de géométrie. J. Math. Pures Appl. 7, 215–216 (1842)

    Google Scholar 

  17. Besicovitch, A.S.: A problem on a circle. J. Lond. Math. Soc. 36, 241–244 (1961)

    MathSciNet  MATH  Google Scholar 

  18. Bianchi, G., Gruber, P.M.: Characterization of ellipsoids. Arch. Math. (Basel) 49, 344–350 (1987)

    MathSciNet  MATH  Google Scholar 

  19. Blaschke, W.: Räumliche Variationsprobleme mit symmetrischen Transversalitätsbedingungen. Ber. Math. Phys. Kl. Königl. Sächs. Ges. Wiss. Leipzig 68, 50–55 (1916)

    Google Scholar 

  20. Blaschke, W.: Kreis und Kugel. Veit, Leipzig (1916)

    MATH  Google Scholar 

  21. Blaschke, W.: Altes und Neues von Ellipse und Ellipsoid. Jahresber. Deutsch. Math. Vereinig. 26, 220–230 (1917)

    MATH  Google Scholar 

  22. Blaschke, W.: Vorlesungen über Differentialgeometrie II. Affine Differentialgeometrie. Springer, Berlin (1923)

    MATH  Google Scholar 

  23. Blaschke, W.: Zur Affingeometrie der Eilinien und Elfächen. Math. Nachr. 15, 258–264 (1956)

    MathSciNet  MATH  Google Scholar 

  24. Blaschke, W., Hessenberg, G.: Lehrsätze über konvexe Körper. Jahresber. Deutsch. Math.-Vereinig. 26, 215–220 (1917)

    MATH  Google Scholar 

  25. Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper. Springer, Berlin, 1934. English translation: Theory of convex bodies. BCS Associates, Moscow, ID (1987)

  26. Borodin, P.A.: Quasi-orthogonal sets and conditions for the Hilbert property of a Banach space. Sb. Math. 188, 1171–1182 (1997)

    MathSciNet  MATH  Google Scholar 

  27. Borodin, P.A.: A new proof of Blaschke’s ellipsoid theorem. Mosc. Univ. Math. Bull. 58(3), 6–10 (2003)

    MathSciNet  MATH  Google Scholar 

  28. Brunn, H.: Über Kurven ohne Wendepunkte. Habilitationschrift, Ackermann, München (1889)

  29. Burton, G.R.: Sections of convex bodies. J. Lond. Math. Soc. 12, 331–336 (1976)

    MathSciNet  MATH  Google Scholar 

  30. Burton, G.R.: On the sum of a zonotope and an ellipsoid. Comment. Math. Helv. 51, 369–387 (1976)

    MathSciNet  MATH  Google Scholar 

  31. Burton, G.R.: Some characterisations of the ellipsoid. Israel J. Math. 28, 339–349 (1977)

    MathSciNet  MATH  Google Scholar 

  32. Burton, G.R.: Congruent sections of a convex body. Pac. J. Math. 81, 303–316 (1979)

    MathSciNet  MATH  Google Scholar 

  33. Burton, G.R., Larman, D.G.: On a problem of J. Höbinger. Geom. Dedicata 5, 31–42 (1976)

    MATH  Google Scholar 

  34. Burton, G.R., Mani, P.: A characterization of the ellipsoid in terms of concurent sections. Comment. Math. Helv. 53, 485–507 (1978)

    MathSciNet  MATH  Google Scholar 

  35. Busemann, H.: The Geometry of Geodesics. Academic Press, New York (1955)

    MATH  Google Scholar 

  36. Busemann, H.: Timelike spaces. Dissertationes Math. (Rozprawy Mat.) 53, p. 52 (1967)

  37. Carathéodory, C.: Über den Variabilitätsbereich der Koefficienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann. 64, 95–115 (1907)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  39. Coolidge, J.L.: A History of the Conic Sections and Quadric Surfaces. Oxford University Press, Oxford (1945)

    MATH  Google Scholar 

  40. Danzer, L.W.: A characterization of the circle. In: Klee, V. (ed.) Convexity, pp. 99–100. American Mathematical Society, Providence (1963)

    Google Scholar 

  41. Düvelmeyer, N.: Convex bodies with equiframed two-dimensional sections. Arch. Math. (Basel) 88, 181–192 (2007)

    MathSciNet  MATH  Google Scholar 

  42. Gardner, R.J.: Geometric Tomography. Cambridge University Press, New York (1995)

    MATH  Google Scholar 

  43. Goodey, P.R.: Homothetic ellipsoids. Math. Proc. Camb. Philos. Soc. 93, 25–34 (1983)

    MathSciNet  MATH  Google Scholar 

  44. Gromov, M.L.: On a geometric hypothesis of Banach. Izv. Akad. Nauk. SSSR Ser. Mat. 31, 1105–1114 (1967)

    MathSciNet  MATH  Google Scholar 

  45. Gruber, P.M.: Über kennzeichende Eigenschaften von eucklidischen Räumen und Ellipsoiden. I. J. Reine Angew. Math. 265, 61–83 (1974)

    MathSciNet  MATH  Google Scholar 

  46. Gruber, P.M.: Über kennzeichnende Eigenschaften von eukclidischen Räumen und Ellipsoiden. II. J. Reine Angew. Math. 270, 123–142 (1974)

    MathSciNet  MATH  Google Scholar 

  47. Gruber, P.M.: Über kennzeichende Eigenschaften von eucklidischen Räumen und Ellipsoiden. III. Monatsh. Math. 78, 311–340 (1974)

    MathSciNet  MATH  Google Scholar 

  48. Gruber, P.M.: A Helmholtz-Lie type characterization of ellipsoids. I. Discrete Comput. Geom. 13, 517–527 (1995)

    MathSciNet  MATH  Google Scholar 

  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)

  50. Gruber, P.M., Ludwig, M.: A Helmholtz–Lie type characterization of ellipsoids. II. Discrete Comput. Geom. 16, 55–67 (1996)

    MathSciNet  MATH  Google Scholar 

  51. Gruber, P.M., Ódor, T.: Ellipsoids are the most symmetric convex bodies. Arch. Math. (Basel) 73, 394–400 (1999)

    MathSciNet  MATH  Google Scholar 

  52. Grünbaum, B.: Arrangements and Spreads. American Mathematical Society, Providence (1972)

    MATH  Google Scholar 

  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. 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)

    Google Scholar 

  55. Helmholtz, H.: Über die Tatsachen, die der Geometrie zu Grunde liegen, pp. 193–221. Göttingen Nachrichten, Göttingen (1868)

    MATH  Google Scholar 

  56. Höbinger, J.: Über einen Satz von Aitchison, Petty und Rogers. Ph.D. Thesis. Techn. Univ. Wien (1974)

  57. Ivanov, B.A.: Straight line segments on the boundary of a convex body. Ukrain. Geom. Sb. No. 13, 69–71 (1973)

    MathSciNet  Google Scholar 

  58. Ivanov, S.: Monochromatic Finsler surfaces and a local ellipsoid characterization. Proc. Am. Math. Soc. 146, 1741–1755 (2018)

    MathSciNet  MATH  Google Scholar 

  59. Jerónimo-Castro, J., McAllister, T.B.: Two characterizations of ellipsoidal cones. J. Convex Anal. 20, 1181–1187 (2013)

    MathSciNet  MATH  Google Scholar 

  60. Kakeya, S.: On some properties of convex curves and surfaces. Tôhoku Math. J. 8, 218–221 (1915)

    MATH  Google Scholar 

  61. Kakutani, S.: Some characterizations of Euclidean space. Jpn. J. Math. 15, 93–97 (1939)

    MathSciNet  MATH  Google Scholar 

  62. Kelly, P., Straus, E.G.: On the projective centres of convex curves. Can. J. Math. 12, 568–581 (1960)

    MathSciNet  MATH  Google Scholar 

  63. Klee, V.L.: Some characterizations of convex polyhedra. Acta Math. 102, 79–107 (1959)

    MathSciNet  MATH  Google Scholar 

  64. Kneser, M.: Eibereiche mit geraden Schwerlinien. Math. Phys. Semesterber. 1, 97–98 (1949)

    MathSciNet  MATH  Google Scholar 

  65. Kojima, T.: On characteristic properties of the conic and quadric. Sci. Rep. Tôhoku Univ. 8, 67–78 (1919)

    MATH  Google Scholar 

  66. Kubota, T.: On the theory of closed convex surface. Proc. Lond. Math. Soc. 14, 230–239 (1914)

    MATH  Google Scholar 

  67. Kubota, T.: Einfache Beweise eines Satzes über die konvexe geschlossene Fläche. Sci. Rep. Tôhoku Univ. 3, 235–255 (1914)

    MATH  Google Scholar 

  68. Kubota, T.: Über die konvexe geschlossene Fläche. Sci. Rep. Tôhoku Univ. 3, 277–287 (1914)

    MATH  Google Scholar 

  69. Kubota, T.: On a characteristic property of the ellipse. Tôhoku Math. J. 9, 148–151 (1916)

    MATH  Google Scholar 

  70. Larman, D.G.: A note on the false centre problem. Mathematika 21, 216–227 (1974)

    MathSciNet  MATH  Google Scholar 

  71. Larman, D., Montejano, L., Morales-Amaya, E.: Characterization of ellipsoids by means of parallel translated sections. Mathematika 56, 363–378 (2010)

    MathSciNet  MATH  Google Scholar 

  72. Lenz, H.: Einige Anwendungen der projektiven Geometrie auf Fragen der Flächentheorie. Math. Nachr. 18, 346–359 (1958)

    MathSciNet  MATH  Google Scholar 

  73. Lie, S.: Theorie der Transformationsgruppen. Bd. III. Teubner, Leipzig (1893)

  74. Lie, S.: Bestimmung aller Flächen, die eine continuirliche Schar von projectiven Transformationen gestatten. Ber. Sächs. Akad. Wiss. Leipzig 47, 209–260 (1895)

    MATH  Google Scholar 

  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)

  76. Mani, P.: Fields of planar bodies tangent to spheres. Monatsh. Math. 74, 145–149 (1970)

    MathSciNet  MATH  Google Scholar 

  77. Marchaud, A.: Sur les ovales. Ann. Soc. Polon. 21, 324–331 (1948)

    MathSciNet  MATH  Google Scholar 

  78. Marchaud, A.: Un théorème sur les corps convexes. Ann. Scient. École Norm. Supér. 76, 283–304 (1959)

    MathSciNet  MATH  Google Scholar 

  79. Martini, H., Soltan, V.: Combiatorial problems on the illumination of convex bodies. Aequ. Math. 57, 121–152 (1999)

    MATH  Google Scholar 

  80. Mauldin, R.D. (ed.): The Scottish Book. Birkäuser, Boston (1981)

    MATH  Google Scholar 

  81. Mazur, S.: Quelques propriétés caractéristiques des espaces euclidiens. C. R. Acad. Sci. Paris 207, 761–764 (1938)

    MATH  Google Scholar 

  82. Montejano, L.: Convex bodies with homothetic sections. Bull. Lond. Math. Soc. 23, 381–386 (1991)

    MathSciNet  MATH  Google Scholar 

  83. Montejano, L., Morales-Amaya, E.: Characterization of ellipsoids and polarity in convex sets. Mathematika 50, 63–72 (2003)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  85. Montejano, L., Morales-Amaya, E.: Shaken false centre theorem. I. Mathematika 54, 41–46 (2007)

    MathSciNet  MATH  Google Scholar 

  86. Nakagawa, S.: On some theorems regarding ellipsoids. Tôhoku Math. J. 8, 11–13 (1915)

    MATH  Google Scholar 

  87. Nakajima, S.: Eilinien mit geraden Schwerlinien. Jpn. J. Math. 5, 81–84 (1928)

    MATH  Google Scholar 

  88. Nakajima, S.: Über konvexe Kurven und Flächen. Tôhoku Math. J. 29, 227–230 (1928)

    MATH  Google Scholar 

  89. Olovjanishnikov, S.P.: On a characterization of the ellipsoid. Učen. Zap. Leningrad. State Univ. Ser. Mat. 83, 114–128 (1941)

    Google Scholar 

  90. Petty, C.M.: Ellipsoids. In: Gruber, P.M., Wills, J.M. (eds.) Convexity and its Applications, pp. 264–276. Birkhäuser, Basel (1983)

    Google Scholar 

  91. Phillips, R.S.: A characterization of Euclidean spaces. Bull. Am. Math. Soc. 46, 930–933 (1940)

    MathSciNet  MATH  Google Scholar 

  92. Rogers, C.A.: Sections and projections of convex bodies. Port. Math. 24, 99–103 (1965)

    MathSciNet  MATH  Google Scholar 

  93. Rudin, W., Smith, K.T.: Linearity of best approximation: a characterization of ellipsoids. Indag. Math. 23, 97–103 (1961)

    MathSciNet  MATH  Google Scholar 

  94. Šaĭdenko, A.V.: Some characteristic properties of an ellipsoid. Sibirsk. Mat. Ž. 21, 232–234 (1980)

    MathSciNet  Google Scholar 

  95. Socié-Méthou, E.: Caractérisation des ellipsoïdes par leurs groupes d’automorphismes. Ann. Sci. École Norm. Sup. 35, 537–548 (2002)

    MathSciNet  MATH  Google Scholar 

  96. Soltan, V.: Convex bodies with polyhedral midhypersurfaces. Arch. Math. 65, 336–341 (1995)

    MathSciNet  MATH  Google Scholar 

  97. Soltan, V.: Affine diameters of convex-bodies—a survey. Expo. Math. 23, 47–63 (2005)

    MathSciNet  MATH  Google Scholar 

  98. Soltan, V.: Convex solids with planar midsurfaces. Proc. Am. Math. Soc. 136, 1071–1081 (2008)

    MathSciNet  MATH  Google Scholar 

  99. Soltan, V.: Convex solids with homothetic sections through given points. J. Convex Anal. 16, 473–486 (2009)

    MathSciNet  MATH  Google Scholar 

  100. Soltan, V.: Convex quadrics. Bul. Acad. Ştiinţe Repub. Moldova. Mat. 3, 94–106 (2010)

    MathSciNet  MATH  Google Scholar 

  101. Soltan, V.: Convex solids with hyperplanar midsurfaces for restricted families of chords. Bul. Acad. Ştiinţe Repub. Moldova. Mat. 2, 23–40 (2011)

    MathSciNet  MATH  Google Scholar 

  102. Soltan, V.: Convex solids with hyperplanar shadow-boundaries. J. Convex Anal. 19, 591–607 (2012)

    MathSciNet  MATH  Google Scholar 

  103. Soltan, V.: Convex solids whose point-source shadow-boundaries lie in hyperplanes. J. Geom. 103, 149–160 (2012)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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. Soltan, V.: Convex hypersurfaces with hyperplanar intersections of their homothetic copies. J. Convex Anal. 22, 145–159 (2015)

    MathSciNet  MATH  Google Scholar 

  107. Soltan, V.: Lectures on Convex Sets. World Scientific, Hackensack (2015)

    MATH  Google Scholar 

  108. Soltan, V.: Convex surfaces with planar polar sets and point-source shadow-boundaries. J. Convex Anal. 24, 645–660 (2017)

    MathSciNet  MATH  Google Scholar 

  109. Süss, W.: Kennzeichnende Eigenschaften der Kugel als Folgerung eines Browerschen Fixpunktsatzes. Comment. Math. Helv. 20, 61–64 (1947)

    MathSciNet  MATH  Google Scholar 

  110. Süss, W.: Eine elementare kennzeichnende Eigenschaft des Ellipsoids. Math.-Phys. Semesterber. 3, 57–58 (1953)

    MathSciNet  MATH  Google Scholar 

  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. Tietze, H.: Über Konvexheit im kleinen und im grossen und über gewisse den Punkten einer Menge zugeordnete Dimensionszahlen. Math. Z. 28, 697–707 (1928)

    MathSciNet  MATH  Google Scholar 

  113. Watson, A.G.D.: On Mizel’s problem. J. Lond. Math. Soc. 37, 307–308 (1962)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA, 22030, USA

    Valeriu Soltan

Authors
  1. Valeriu Soltan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Valeriu Soltan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Soltan, V. Characteristic properties of ellipsoids and convex quadrics. Aequat. Math. 93, 371–413 (2019). https://doi.org/10.1007/s00010-018-0620-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-018-0620-1

Keywords

Mathematics Subject Classification

Search

Navigation