Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Geometric convexity. III. Embedding


Authors: John Cantwell and David C. Kay
Journal: Trans. Amer. Math. Soc. 246 (1978), 211-230
MSC: Primary 52A01
DOI: https://doi.org/10.1090/S0002-9947-1978-0515537-7
MathSciNet review: 515537
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The straight line spaces of dimension three or higher which were considered by the first author in previous papers are shown to be isomorphic with a strongly open convex subset of a real vector space. To achieve this result we consider the classical descriptive geometry studied in various papers and textbooks by Pasch, Hilbert, Veblen, Whitehead, Coxeter, Robinson, and others, with the significant difference that the geometry considered here is not restricted to 3 dimensions. Our main theorem (which is well known in dimension 3) is that any such geometry is isomorphic to a strongly open convex subset of a real vector space whose ``chords'' play the role of lines.


References [Enhancements On Off] (What's this?)

  • [1] E. Artin, Geometrie algebra, Interscience, New York, 1957. MR 0082463 (18:553e)
  • [2] G. Birkhoff, Lattice theory, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1948. MR 0029876 (10:673a)
  • [3] V. W. Bryant and R. J. Webster, Convexity spaces. I; The basic properties, J. Math. Anal. Appl. 37 (1972), 206-213. MR 0296812 (45:5871)
  • [4] -, Convexity spaces. II; Separation, J. Math. Anal. Appl. 43 (1973), 321-327. MR 0380622 (52:1519)
  • [5] H. Busemann, The geometry of geodesies, Academic Press, New York, 1955. MR 0075623 (17:779a)
  • [6] J. Cantwell, Convexity in straight line spaces, in Proc. Conf. Convexity and Combinatorial Geometry, Univ. of Oklahoma, Norman, Oklahoma, 1971, pp. 65-102. MR 0338932 (49:3695)
  • [7] -, Geometric convexity. I, Bull. Inst. Math. Acad. Sinica 2 (1974), 289-307. MR 0358567 (50:11027)
  • [8] -, Geometric convexity. II: Topology (preprint).
  • [9] H. S. M. Coxeter, Introduction to geometry, Wiley, New York, 1961. MR 0123930 (23:A1251)
  • [10] -, Non-Euclidean geometry, Univ. of Toronto Press, Toronto, 1942.
  • [11] J.-P. Doignon, Caractérisations d'espaces de Pasch-Peano, Acad. Roy. Belg. Cl. Sci. (5) 62 (1976), 679-699. MR 0470849 (57:10593)
  • [12] L. Fuchs, Partially ordered algebraic systems, Pergamon, London, 1963. MR 0171864 (30:2090)
  • [13] R. Hartshorne, Foundations of projective geometry, Harvard Univ. Lecture Notes, Benjamin, New York, 1967. MR 0222751 (36:5801)
  • [14] D. C. Kay and W. Meyer, A convexity structure admits but one real linearization of dimension greater than one, J. London Math. Soc. (2) 7 (1973), 124-130. MR 0322477 (48:839)
  • [15] D. C. Kay and E. W. Womble, Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers, Pacific J. Math. 38 (1971), 471-485. MR 0310766 (46:9864)
  • [16] P. Mah, S. A. Naimpally and J. H. M. Whitfield, Linearization of a convexity space, J. London Math. Soc. (2) 13 (1976), 209-214. MR 0415510 (54:3596)
  • [17] F. R. Moulton, A simple non-Desarguesian projective geometry, Trans. Amer. Math. Soc. 3 (1902), 192-195. MR 1500595
  • [18] W. Nitka, On convex metric spaces. II, Fund. Math. 72 (1971), 115-129. MR 0317289 (47:5836)
  • [19] E. Nunnally, A metric characterization of cells, Trans. Amer. Math. Soc. 184 (1973), 317-325. MR 0326693 (48:5036)
  • [20] G. de B. Robinson, Foundations of geometry, Univ. of Toronto Press, Toronto, 1940. MR 0002929 (2:135a)
  • [21] G. Š. Rubinsteĭn, On an intrinsic characteristic of relatively open convex sets, Soviet Math. Dokl. 11 (1970), 1076-1079.
  • [22] O. Veblen, A system of axioms for geometry, Trans. Amer. Math. Soc. 5 (1904), 343-384. MR 1500678
  • [23] A. N. Whitehead, The axioms of descriptive geometry, Hafner, New York, 1907.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 52A01

Retrieve articles in all journals with MSC: 52A01


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0515537-7
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society