# A dual proof for the linearization of the convexity spaces

## 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

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

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## Paper coordinates

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

MR: 86g:52004