A Characterization of Minkowskian Geometry

Menger called a finitely-compact metric space in which any two distinct points are contained in exactly one set congruent to a euclidean straight line a straight-line space (S.L. space) [ l ] . 1 He observed that in such spaces a segment (or vector) addition can be defined in a very simple way (see §2 of the present paper) which satisfies all requirements for an abelian group except, in general, the associative law. Menger then put the question of determining those geometries in which the associative law holds. I t is the purpose of the present note to show that the Minkowskian geometries furnish the answer. 1. Straight-line spaces. By a straight-line (S.L.) in a metric space R is understood a continuous curve which may be mapped congruently onto the real axis, tha t is, which admits of a parametrization P(r), — oo < r < oo, with P ( T I ) P ( T 2 ) = | ri — r2 | . The numbers r are the isometric coordinates of the points of the S.L. A segment joining two points P(r i) , <2(2)CP, written (PQ), is a congruent image of the closed interval between ri and r2. An S.L. space is one in which any pair of its points X, F is contained in a unique S.L. [ I F ] . The following conditions are sufficient for an S.L. space: the space is (1) metric, (2) finitely-compact, (3) convex, (4) externally convex, (5) if (XFZi), (XYZ2), and YZx=YZ2l then Zi = Z2. We shall suppose, hereafter, that these conditions hold. If X„->X, Yn-*Y and X^ Y then [ l n F n ] > [ l F ] . According to Menger M is an internal center of X and F, written M(X, F), if (XMY) and XM=MY. Also, if (EXY) and £ X = X F , E is an external center of X and F, written E(X, Y) or (F , X)E. In an S.L. space every point-pair has a unique internal center and two external centers.