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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