A dual proof for the linearization of the convexity spaces


We consider a convexity spaces \(X\) in the sense of V.W. Bryant and R.J. Webster [1]. In [8] we have proved that for such a space the incidence and order Hilbert’s axiom adapted for an arbitrary dimension are satisfied. Thus the axioms I-VIII of O. Veblen [10] hold. If \(\dim X\geq 3\) then the axiom IX also holds. The axiom X is equivalent with \(\dim X\leq3\) and if \(X\) is complete then as a consequence the axiom XI holds.

Therefore is \(X\) is complete, of dimension 3 and satisfies the parallel postulate then all of Veblen’s axioms (I-XII) hold. In this case it is well Known [10] that there exists a real linearization of \(X\); this means that \(X\) can be organized as a real linear space such that its convexity structure be that indebted to the algebraic structure.

J.P. Doignon [5] and J. Cantwell, D.C. Kay [4] have proved the existence of a real linearization for any complete convexity space of dimension \(\geq3\) satisfying the parallel postulate; the linearization is unique up to a translation of the origin according to a theorems of D.C. Kay, W. Meyer [6].

In our paper this result will be proved from a different point of view using a more general result of P. Man, S.A. Naimpally, J.H.M. Whitfield [7].

We can summarize our proof as follows: First we solve the case \(3\leq n=\dim X<\infty\) by mathematical induction after \(n\).

In the case \(\dim X=\infty\) we represent \(X\) by \(Y\times L\), where \(Y\subset X\) is a hyperplane of \(X\) and \(L\subset X\) is a line meeting \(Y\), identified with with \(\mathbb{R}\). We denote by \(Y^{\ast}\) the dual of \(Y\) consisting of all functions \(f\) defined on \(Y\) and with values in \(\mathbb{R\equiv L}\) having the graph a hyperplane in \(Y\times L=X\) and satisfying \(f\left( \theta\right) =0\), where \(\theta=Y\cap L\).


Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania



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R. Precup, A dual proof for the linearization of the convexity spaces, Babeş-Bolyai Univ., Faculty of Math., Research Sem. 2 (1983), Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-1983), 119-128


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MR: 86g:52004

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[1] V.W. Bryant, R.J. Webster,  Convexity spaces,  I; The basic properties, J. Math. Anal. Appl., 31, 206-213(1972).
[2] V.W. Bryant, R.J./ Webster,  Convexity spaces, II; Separation, J. Math. Anal. Appl. 43, 321-327 (1973).
[3] V.W. Bryant, R.J. Webster,  Convexity spaces, III; Dimension, J. Math. Anal. Appl., 57, 382-392 (1977).
[4] J. Cantwell, D.C. Kay, Geometrie convexity, III; Embedding, Trans. Amer. Math. Soc. 246, 211-230 (1978).
[5] J.P. Doignon,  Caracteristions d’espaces de Pasch-Peano, Acad. Roy. Belg. Cl. Sci., (5( 62, 679-699 (1976).
[6] D.C. Kay, J. Meyer,  A convexity structure admits but one real liniarization of dimension greater than one,  J. London Math. Soc., (2) 7, 124-130 (1973).
[7] P. Mah, S.A. Naimpally, J.H.M. Whitfield,  Linearization of a  convexity space,  J. London Math. Soc. (2) 13, 209-214 (1976).
[8] R. Precup,  Sur l’approximaiton des espaces a convexite, L’Anal. Num. Th. L’Approx. 9, 2, 255-260 (1980).
[9] R. Precup, Functii definite pe spatii de convexitate; spatii duale de conv exitate, Lucr. Sem. Itin. Ec. Funct., Aprox., Conv., Cluj-Napoca 16-17 mai (1980).
[10] O.Veblen, A system of axioms for geometry, Trans. Amer. Math. Soc. 5, 343-384(1904).

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