Abstract
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 \(\theta=Y\cap L\).
Authors
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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