| Title:
|
A new type of orthogonality for normed planes (English) |
| Author:
|
Martini, Horst |
| Author:
|
Spirova, Margarita |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
339-349 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $d\geq 3$. (English) |
| Keyword:
|
chordal orthogonality |
| Keyword:
|
Feuerbach circle |
| Keyword:
|
inner product space |
| Keyword:
|
James orthogonality |
| Keyword:
|
Minkowski plane |
| Keyword:
|
normed linear space |
| Keyword:
|
normed plane |
| Keyword:
|
orthocentricity |
| Keyword:
|
Wallace line |
| MSC:
|
46B20 |
| MSC:
|
46C15 |
| MSC:
|
52A21 |
| idZBL:
|
Zbl 1224.46026 |
| idMR:
|
MR2657953 |
| . |
| Date available:
|
2010-07-20T16:42:07Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140573 |
| . |
| Reference:
|
[1] Alonso, J., Martini, H., Mustafaev, Z.: On orthogonal chords in normed planes.(to appear) in Rocky Mountains J. Math. MR 2845932 |
| Reference:
|
[2] Amir, D.: Characterizations of Inner Product Spaces.Birkhäuser Verlag, Basel-Boston-Stuttgart (1986). Zbl 0617.46030, MR 0897527 |
| Reference:
|
[3] Asplund, E., Grünbaum, B.: On the geometry of Minkowski planes.Enseign. Math. 6 (1960), 299-306. MR 0137023 |
| Reference:
|
[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. MR 1057208 |
| Reference:
|
[5] Birkhoff, G.: Orthogonality in linear metric spaces.Duke Math. J. 1 (1935), 169-172 \JFM 61.0634.01. Zbl 0012.30604, MR 1545873, 10.1215/S0012-7094-35-00115-6 |
| Reference:
|
[6] Hofmann, J. E.: Zur elementaren Dreiecksgeometrie in der komplexen Ebene.Enseign. Math. 4 (1958), 178-211. Zbl 0085.14803, MR 0102768 |
| Reference:
|
[7] James, R. C.: Orthogonality in normed linear spaces.Duke Math. J. 12 (1945), 291-301. Zbl 0060.26202, MR 0012199, 10.1215/S0012-7094-45-01223-3 |
| Reference:
|
[8] James, R. C.: Inner product in normed linear spaces.Bull. Amer. Math. Soc. 53 (1947), 559-566. MR 0021242, 10.1090/S0002-9904-1947-08831-5 |
| Reference:
|
[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. Zbl 0888.51018, MR 1470989 |
| Reference:
|
[10] Martini, H., Spirova, M.: The Feuerbach circle and orthocentricity in normed planes.Enseign. Math. 53 (2007), 237-258. MR 2455944 |
| Reference:
|
[11] Martini, H., Spirova, M.: Clifford's chain of theorems in strictly convex Minkowski planes.Publ. Math. Debrecen 72 (2008), 371-383. Zbl 1174.51005, MR 2406927 |
| Reference:
|
[12] Martini, H., Swanepoel, K. J., Weiss, G.: The geometry of Minkowski spaces-a survey, Part I.Expositiones Math. 19 (2001), 97-142. MR 1835964, 10.1016/S0723-0869(01)80025-6 |
| Reference:
|
[13] Thompson, A. C.: Minkowski Geometry.Encyclopedia of Mathematics and its Applications, Vol. 63, Cambridge University Press, Cambridge (1996). Zbl 0868.52001, MR 1406315 |
| Reference:
|
[14] Weiss, G.: The concepts of triangle orthocenters in Minkowski planes.J. Geom. 74 (2002), 145-156. Zbl 1028.51005, MR 1940598, 10.1007/PL00012533 |
| . |
