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Characterizations of Convex Quadrics in Terms of Plane Quadric Sections, Midsurfaces, and Shadow-Boundaries

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New Frontiers of Multidisciplinary Research in STEAM-H (Science, Technology, Engineering, Agriculture, Mathematics, and Health)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 90))

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Abstract

It is well known that the middle points of any family of parallel chords of a real quadric surface Q in the Euclidean space Rn belong to a hyperplane, and that a similar property holds for the shadow-boundaries of Q. In this article we review the existing results and add some new ones which characterize convex quadrics among convex hypersurfaces in Rn, possibly unbounded, in terms of plane quadric sections, hyperplanarity of their midsurfaces and shadow-boundaries.

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References

  1. Alexandrov, A. D. (1939). On convex surfaces with plane shadow-boundaries. (Russian) Matematicheskii Sbornik, 5, 309–316.

    Google Scholar 

  2. Amir, D. (1986). Characterizations of inner product spaces. Basel: Birkhäuser.

    Book  MATH  Google Scholar 

  3. Berger, K. H. (1936). Eilinien mit perspektiv liegenden Tangenten-und Sehnendreiecken (vol. 4, pp. 1–11). S.-B. Heidelberg: Akad. Wiss.

    Google Scholar 

  4. Bertrand, J. (1842). Démonstration d’un théoreme de géométrie. Journal de Mathématiques Pures et Appliquées, 7, 215–216.

    Google Scholar 

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

    Google Scholar 

  6. Blaschke, W. (1916). Kreis und Kugel. Leipzig: Viet.

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. Blaschke, W. (1956). Zur Affingeometrie der Eilinien und Elfächen. Mathematische Nachrichten, 15, 258–264.

    Article  MathSciNet  MATH  Google Scholar 

  9. Blaschke, W., & Hessenberg, G. (1917). Lehrsätze über konvexe Körper. Jahresber. Deutsche Mathematiker-Vereinigung, 26, 215–220.

    Google Scholar 

  10. Borodin, P. A. (1997). Quasi-orthogonal sets and conditions for the Hilbert property of a Banach space. (Russian) Matematicheskii Sbornik, 188, 63–74. English translation in: Matematicheskii Sbornik, 188, 1171–1182.

    Google Scholar 

  11. Borodin, P. A. (2003). A new proof of Blaschke’s ellipsoid theorem. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh., 3, 17–22. English translation in: Moscow University Mathematics Bulletin, 58(3), 6–10.

    Google Scholar 

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

    Google Scholar 

  13. Burton, G. R. (1976). On the sum of a zonotope and an ellipsoid. Commentarii Mathematici Helvetici, 51, 369–387.

    Article  MathSciNet  MATH  Google Scholar 

  14. Busemann, H. (1955). The geometry of geodesics. New York: Academic Press.

    MATH  Google Scholar 

  15. Chakerian, G. D. (1965). The affine image of a convex body of constant breadth. Israel Journal of Mathematics, 3, 19–22.

    Article  MathSciNet  MATH  Google Scholar 

  16. Gardner, R. J. (1995). Geometric tomography. New York: Cambridge University Press.

    MATH  Google Scholar 

  17. Grinberg, E. L. (1991). Isoperimetric inequalities for k-dimensional cross-sections and projections of convex bodies. Mathematische Annalen, 291, 75–87.

    Article  MathSciNet  Google Scholar 

  18. Gruber, P. M. (1974). Über kennzeichende Eigenschaften von eucklidischen Raumen und Ellipsoiden. I. Journal für die reine und Angewandte Mathematik, 265, 61–83.

    Google Scholar 

  19. Gruber, P. M. (1974). Über kennzeichende Eigenschaften von eucklidischen Raumen und Ellipsoiden. III. Monatshefte für Mathematik, 78, 311–340.

    Google Scholar 

  20. Grünbaum, B. (1972). Arrangements and spreads. Conference Board of the Mathematical Sciences, No. 10. Providence, RI: American Mathematical Society.

    Google Scholar 

  21. Jitomirskii, O. K. (1938). On surfaces with plane shadow-boundaries. (Russian) Matematicheskii Sbornik, 3, 347–352.

    Google Scholar 

  22. Kakutani, S. (1939). Some characterizations of Euclidean space. Japanese Journal of Mathematics, 15, 93–97.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  24. Kubota, T. (1916). On a characteristic property of the ellipse. Tohoku Mathematical Journal, 9, 148–151.

    MATH  Google Scholar 

  25. Lenz, H. (1958). Einige Anwendungen der projektiven Geometrie auf Fragen der Flächentheorie. Mathematische Nachrichten, 18, 346–359.

    Article  MathSciNet  MATH  Google Scholar 

  26. Marchaud, A. (1959). Un théoréme sur les corps convexes. Annales Scientifiques de lÉcole Normale Supérieure, 76, 283–304.

    MathSciNet  MATH  Google Scholar 

  27. Martini, H., & Soltan, V. (1999). Combiatorial problems on the illumination of convex bodies. Aequationes Mathematicae, 57, 121–152.

    Article  MathSciNet  MATH  Google Scholar 

  28. Montejano, L., & Morales-Amaya, E. (2007). Variations of classic characterizations of ellipsoids and a short proof of the false centre theorem. Mathematika, 54, 35–40.

    Article  MathSciNet  MATH  Google Scholar 

  29. Nakajima, S. (1928). Eilinien mit geraden Schwerlinien. Japanese Journal of Mathematics, 5, 81–84.

    MATH  Google Scholar 

  30. Phillips, R. S. (1940). A characterization of Euclidean spaces. Bulletin of the American Mathematical Society, 46, 930–933.

    Article  MathSciNet  Google Scholar 

  31. Rudin, W., & Smith, K. T. (1961). Linearity of best approximation: a characterization of ellipsoids. Indagationes Mathematicae, 23, 97–103.

    MathSciNet  Google Scholar 

  32. Šaĭdenko, A. V. (1980). Some characteristic properties of an ellipsoid. (Russian) Sibirskii Matematicheskii Zhurnal, 21, 232–234.

    Google Scholar 

  33. Schwenk, A. (1985). Affinsphären mit ebenen schattengrenzen. Lecture Notes in Math. vol. 1156, pp. 296–315. Berlin: Springer.

    Google Scholar 

  34. Schneider, R. (1967). Zur affinen Differentialgeometrie im Grossen. I. Mathematische Zeitschrift, 101, 375–406.

    Google Scholar 

  35. Soltan, V. (1995). Convex bodies with polyhedral midhypersurfaces. Archiv der Mathematik, 65, 336–341.

    Article  MathSciNet  MATH  Google Scholar 

  36. Soltan, V. (2005). Affine diameters of convex bodies–a survey. Expositiones Mathematicae, 23, 47–63.

    Article  MathSciNet  MATH  Google Scholar 

  37. Soltan, V. (2008). Convex solids with planar midsurfaces. Proceedings of the American Mathematical Society, 136, 1071–1081.

    Google Scholar 

  38. Soltan, V. (2009). Convex solids with homothetic sections through given points. Journal of Convex Analysis, 16, 473–486.

    MathSciNet  MATH  Google Scholar 

  39. Soltan, V. (2010). Convex quadrics. Buletinul Academiei de Stiinte a Republicii Moldova Matematica, 3, 94–106.

    MathSciNet  Google Scholar 

  40. Soltan, V. (2011). Convex solids with hyperplanar midsurfaces for restricted families of chords. Buletinul Academiei de Stiinte a Republicii Moldova Matematica, 2, 23–40.

    MathSciNet  Google Scholar 

  41. Soltan, V. (2012). Convex solids with hyperplanar shadow-boundaries. Journal of Convex Analysis, 19, 591–607.

    MathSciNet  MATH  Google Scholar 

  42. Soltan, V. (2012). Convex quadrics and their characterizations by means of plane sections. In B. Toni, et al. (Eds.), Bridging mathematics, statistics, engineering and technology (pp. 131–145). Springer Proceedings in Mathematics & Statistics, vol. 24.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  44. Süss, W., Viet, U., & Berger, K. H. (1971). Konvexe Figuren. In H. Behnke, K. Fladt, & H. Kunle (Eds.) Grundzüge der mathematik. Bd. II: Geometrie. Teil B, (pp. 361–381). Göttingen: Vanderhoeck and Ruprecht.

    Google Scholar 

  45. Thompson, A. C. (1996). Minkowski geometry. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  46. Webster, R. J. (1994). Convexity. Oxford: Oxford University Press.

    MATH  Google Scholar 

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Acknowledgements

The authors thanks the referee for helpful comments on an earlier draft of this article.

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  1. George Mason University, Fairfax, VA, 22030, USA

    Valeriu Soltan

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  1. Valeriu Soltan
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Correspondence to Valeriu Soltan .

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  1. Dept of Mathematics & Computer Sciences, Virginia State University, Petersburg, Virginia, USA

    Bourama Toni

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Soltan, V. (2014). Characterizations of Convex Quadrics in Terms of Plane Quadric Sections, Midsurfaces, and Shadow-Boundaries. In: Toni, B. (eds) New Frontiers of Multidisciplinary Research in STEAM-H (Science, Technology, Engineering, Agriculture, Mathematics, and Health). Springer Proceedings in Mathematics & Statistics, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-07755-0_4

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