A new type of orthogonality for normed planes

Horst Martini; Margarita Spirova

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 2, page 339-349
  • ISSN: 0011-4642

Abstract

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In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions d ≥ 3 .

How to cite

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Martini, Horst, and Spirova, Margarita. "A new type of orthogonality for normed planes." Czechoslovak Mathematical Journal 60.2 (2010): 339-349. <http://eudml.org/doc/38011>.

@article{Martini2010,
abstract = {In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions $d\ge 3$.},
author = {Martini, Horst, Spirova, Margarita},
journal = {Czechoslovak Mathematical Journal},
keywords = {chordal orthogonality; Feuerbach circle; inner product space; James orthogonality; Minkowski plane; normed linear space; normed plane; orthocentricity; Wallace line; chordal orthogonality; Feuerbach circle; inner product space; James orthogonality; Minkowski plane; normed linear space; normed plane; orthocentricity; Wallace line},
language = {eng},
number = {2},
pages = {339-349},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A new type of orthogonality for normed planes},
url = {http://eudml.org/doc/38011},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Martini, Horst
AU - Spirova, Margarita
TI - A new type of orthogonality for normed planes
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 2
SP - 339
EP - 349
AB - In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions $d\ge 3$.
LA - eng
KW - chordal orthogonality; Feuerbach circle; inner product space; James orthogonality; Minkowski plane; normed linear space; normed plane; orthocentricity; Wallace line; chordal orthogonality; Feuerbach circle; inner product space; James orthogonality; Minkowski plane; normed linear space; normed plane; orthocentricity; Wallace line
UR - http://eudml.org/doc/38011
ER -

References

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  1. Alonso, J., Martini, H., Mustafaev, Z., On orthogonal chords in normed planes, (to appear) in Rocky Mountains J. Math. MR2845932
  2. Amir, D., Characterizations of Inner Product Spaces, Birkhäuser Verlag, Basel-Boston-Stuttgart (1986). (1986) Zbl0617.46030MR0897527
  3. Asplund, E., Grünbaum, B., On the geometry of Minkowski planes, Enseign. Math. 6 (1960), 299-306. (1960) MR0137023
  4. Benitez, C., Orthogonality in normed linear spaces: classification of the different concepts and some open problems, Revista Mat. Univ. Compl. Madrid 2 (1989), 53-57. (1989) MR1057208
  5. Birkhoff, G., 10.1215/S0012-7094-35-00115-6, Duke Math. J. 1 (1935), 169-172 1.0634.01. (1935) Zbl0012.30604MR1545873DOI10.1215/S0012-7094-35-00115-6
  6. Hofmann, J. E., Zur elementaren Dreiecksgeometrie in der komplexen Ebene, Enseign. Math. 4 (1958), 178-211. (1958) Zbl0085.14803MR0102768
  7. James, R. C., 10.1215/S0012-7094-45-01223-3, Duke Math. J. 12 (1945), 291-301. (1945) Zbl0060.26202MR0012199DOI10.1215/S0012-7094-45-01223-3
  8. James, R. C., 10.1090/S0002-9904-1947-08831-5, Bull. Amer. Math. Soc. 53 (1947), 559-566. (1947) MR0021242DOI10.1090/S0002-9904-1947-08831-5
  9. Martini, H., The three-circles theorem, Clifford configurations, and equilateral zonotopes, Proc. 4th Internat. Congr. Geometry (Thessaloniki, 1996) N. K. Artémiadis and N. K. Stephanidis, Thessaloniki (1997), 281-292. (1997) Zbl0888.51018MR1470989
  10. Martini, H., Spirova, M., The Feuerbach circle and orthocentricity in normed planes, Enseign. Math. 53 (2007), 237-258. (2007) MR2455944
  11. Martini, H., Spirova, M., Clifford's chain of theorems in strictly convex Minkowski planes, Publ. Math. Debrecen 72 (2008), 371-383. (2008) Zbl1174.51005MR2406927
  12. Martini, H., Swanepoel, K. J., Weiss, G., 10.1016/S0723-0869(01)80025-6, Expositiones Math. 19 (2001), 97-142. (2001) MR1835964DOI10.1016/S0723-0869(01)80025-6
  13. Thompson, A. C., Minkowski Geometry, Encyclopedia of Mathematics and its Applications, Vol. 63, Cambridge University Press, Cambridge (1996). (1996) Zbl0868.52001MR1406315
  14. Weiss, G., 10.1007/PL00012533, J. Geom. 74 (2002), 145-156. (2002) Zbl1028.51005MR1940598DOI10.1007/PL00012533

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