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A Convexity Structure Admits But One Real Linearization of Dimension Greater Than One
Author(s) -
Meyer Walter,
Kay David C.
Publication year - 1973
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-7.1.124
Subject(s) - dimension (graph theory) , citation , convexity , computer science , mathematical economics , combinatorics , mathematics , library science , financial economics , economics
If V is a vector space over an ordered field F and has algebraic operations + and o, these algebraic operations determine which subsets of V are convex. Now consider a set V with no algebraic structure, but with a convexity structure, that is, a family # of subsets of V which are closed under intersection. The determination of a linear structure (F, +, o) for V which makes V a vector space over a field F whose convex sets are precisely the members of # from properties of # alone has been called the linearization problem for convexity, abstractly analogous to the metrization problem in topology (see [4] where the problem is solved in a very special case). The linearization is called real if F is the field of real numbers. The purpose of this paper is to show that if a convexity structure has a real linearization of dimension greater than one, it will be unique up to a translation of the origin. Thus, the problem of uniqueness (non-existent for the metrization problem in topology) has been essentially solved, while the problem of existence essentially remains.