Geometric convexity. III. Embedding

Cantwell, John; Kay, David C.

doi:10.1090/S0002-9947-1978-0515537-7

Geometric convexity. III. Embedding
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by John Cantwell and David C. Kay PDF

Trans. Amer. Math. Soc. 246 (1978), 211-230 Request permission

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

E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. MR 0082463
Garrett Birkhoff, Lattice Theory, Revised edition, American Mathematical Society Colloquium Publications, Vol. 25, American Mathematical Society, New York, N. Y., 1948. MR 0029876
V. W. Bryant and R. J. Webster, Convexity spaces. I. The basic properties, J. Math. Anal. Appl. 37 (1972), 206–213. MR 296812, DOI 10.1016/0022-247X(72)90268-5
V. W. Bryant and R. J. Webster, Convexity spaces. II. Separation, J. Math. Anal. Appl. 43 (1973), 321–327. MR 380622, DOI 10.1016/0022-247X(73)90076-0
Herbert Busemann, The geometry of geodesics, Academic Press, Inc., New York, N.Y., 1955. MR 0075623
John Cantwell, Convexity in straight line spaces, Proceedings of the Conference on Convexity and Combinatorial Geometry (Univ. Oklahoma, Norman, Okla., 1971) Dept. Math., Univ. Oklahoma, Norman, Okla., 1971, pp. 65–102. MR 0338932
John Cantwell, Geometric convexity, I, Bull. Inst. Math. Acad. Sinica 2 (1974), 289–307. MR 358567

Geometric convexity

Topology

H. S. M. Coxeter, Introduction to geometry, John Wiley & Sons, Inc., New York-London, 1961. MR 0123930

Non-Euclidean geometry

J.-P. Doignon, Caractérisations d’espaces de Pasch-Peano, Acad. Roy. Belg. Bull. Cl. Sci. (5) 62 (1976), no. 9, 679–699 (French, with English summary). MR 470849, DOI 10.3406/barb.1976.58135
L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR 0171864
Robin Hartshorne, Foundations of projective geometry, W. A. Benjamin, Inc., New York, 1967. Lecture Notes, Harvard University, 1966/67. MR 0222751
Walter Meyer and David C. Kay, A convexity structure admits but one real linearization of dimension greater than one, J. London Math. Soc. (2) 7 (1973), 124–130. MR 322477, DOI 10.1112/jlms/s2-7.1.124
David C. Kay and Eugene W. Womble, Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers, Pacific J. Math. 38 (1971), 471–485. MR 310766, DOI 10.2140/pjm.1971.38.471
P. Mah, S. A. Naimpally, and J. H. M. Whitfield, Linearization of a convexity space, J. London Math. Soc. (2) 13 (1976), no. 2, 209–214. MR 415510, DOI 10.1112/jlms/s2-13.2.209
Forest Ray Moulton, A simple non-Desarguesian plane geometry, Trans. Amer. Math. Soc. 3 (1902), no. 2, 192–195. MR 1500595, DOI 10.1090/S0002-9947-1902-1500595-3
W. Nitka, On convex metric spaces. II, Fund. Math. 72 (1971), no. 2, 115–129. MR 317289, DOI 10.4064/fm-72-2-115-129
Ellard Nunnally, A metric characterization of cells, Trans. Amer. Math. Soc. 184 (1973), 317–325. MR 326693, DOI 10.1090/S0002-9947-1973-0326693-3
Gilbert de B. Robinson, The Foundations of Geometry, Mathematical Expositions, No. 1, University of Toronto Press, Toronto, Ont., 1940. MR 0002929

On an intrinsic characteristic of relatively open convex sets

11

Oswald Veblen, A system of axioms for geometry, Trans. Amer. Math. Soc. 5 (1904), no. 3, 343–384. MR 1500678, DOI 10.1090/S0002-9947-1904-1500678-X

The axioms of descriptive geometry