Authors: Horst Martini and Senlin Wu
View More View Less
  • 1 Chemnitz University of Technology Faculty of Mathematics 09107 Chemnitz Germany
  • 2 Harbin University of Science and Technology Department of Applied Mathematics 150080 Harbin China
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

We give a geometric characterization of inner product spaces among all finite dimensional real Banach spaces via concurrent chords of their spheres. Namely, let x be an arbitrary interior point of a ball of a finite dimensional normed linear space X. If the locus of the midpoints of all chords of that ball passing through x is a homothetical copy of the unit sphere of X, then the space X is Euclidean. Two further characterizations of the Euclidean case are given by considering parallel chords of 2-sections through the midpoints of balls.

  • Alonso, J. and Benítez, C., Orthogonality in normed linear spaces: a survey. Part I: main properties, Extracta Math., 3 (1988), no. 1, 1–15. MR 91e:46021a

    Benítez C. , 'Orthogonality in normed linear spaces: a survey. Part I: main properties ' (1988 ) 3 Extracta Math. : 1 -15.

    • Search Google Scholar
  • Alonso, J. and Benítez, C., Orthogonality in normed linear spaces: a survey. Part II: relations between main orthogonalities, Extracta Math., 4 (1989), no. 3, 121–131. MR 91e:46021b

    Benítez C. , 'Orthogonality in normed linear spaces: a survey. Part II: relations between main orthogonalities ' (1989 ) 4 Extracta Math. : 121 -131.

    • Search Google Scholar
  • Amir, D., Characterizations of Inner Product Spaces, Birkhäuser, Operator Theory: Advances and Applications, 20. Basel, 1986. MR 88m:46001

    Amir D. , '', in Characterizations of Inner Product Spaces , (1986 ) -.

  • Birkhoff, G., Orthogonality in linear metric spaces, Duke Math. J., 1 (1935), no. 2, 169–172. MR 1545873

    Birkhoff G. , 'Orthogonality in linear metric spaces ' (1935 ) 1 Duke Math. J. : 169 -172.

  • Düvelmeyer, N., Angle measures and bisectors in Minkowski planes, Canad. Math. Bull., 48 (2005), no. 4, 523–534. MR 2006g:52008

    Düvelmeyer N. , 'Angle measures and bisectors in Minkowski planes ' (2005 ) 48 Canad. Math. Bull. : 523 -534.

    • Search Google Scholar
  • James, R. C., Orthogonality in normed linear spaces, Duke Math. J., 12 (1945), 291–301. MR 6,273d

    James R. C. , 'Orthogonality in normed linear spaces ' (1945 ) 12 Duke Math. J. : 291 -301.

  • James, R. C., Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc., 61 (1947), 265–292. MR 9,42c

    James R. C. , 'Orthogonality and linear functionals in normed linear spaces ' (1947 ) 61 Trans. Amer. Math. Soc. : 265 -292.

    • Search Google Scholar
  • Martini, H. and Swanepoel, K. J., The geometry of Minkowski spaces — a survey. Part II, Expositiones Math., 22 (2004), no. 2, 93–144. MR 2005h:46028

    Swanepoel K. J. , 'The geometry of Minkowski spaces — a survey. Part II ' (2004 ) 22 Expositiones Math. : 93 -144.

    • Search Google Scholar
  • Martini, H. and Swanepoel, K. J., Antinorms and Radon curves, Aequationes Math., 72 (2006), no. 1–2, 110–138. MR 2007f:52001

    Swanepoel K. J. , 'Antinorms and Radon curves ' (2006 ) 72 Aequationes Math. : 110 -138.

  • Martini, H., Swanepoel, K. J. and Weiss, G., The geometry of Minkowski spaces — a survey. Part I, Expositiones Math., 19 (2001), no. 2, 97–142. MR 2002h:46015a

    Weiss G. , 'The geometry of Minkowski spaces — a survey. Part I ' (2001 ) 19 Expositiones Math. : 97 -142.

    • Search Google Scholar
  • Martini, H. and Senlin Wu, Radial projections of bisectors in Minkowski spaces, Extracta Math., 23 (2008), no. 1, 7–28. MR 2449992

    Senlin W. , 'Radial projections of bisectors in Minkowski spaces ' (2008 ) 23 Extracta Math. : 7 -28.

    • Search Google Scholar
  • Thompson, A. C., Minkowski Geometry, Encyclopedia of Mathematics and Its Applications, Vol. 63, Cambridge University Press, Cambridge, 1996. MR 97f:52001

    Thompson A. C. , '', in Minkowski Geometry , (1996 ) -.