BEALS, R., and D. H. KRANTZ Math. Zeitschr. 101,285--298 (1967) Metrics and Geodesics Induced by Order Relations RICnAgD BEALS and DAVID H. KRANTZ * Received December 27, 1966 Introduction A metric on a space S gives an order on S x S, or equivalently, a mapping of S • S onto a totally ordered set P. Conversely, under suitable conditions such an order induces a uniformity on S which is metrizable. This paper is concerned with the more delicate geometric question of what conditions such an order must satisfy to induce a "canonical" metric on S. By this we mean first that the order induced by the metric coincide with the original order, at least in the small. Second, the ternary relation (xyz) on a metric space (S, d), meaning that d(x, y)+d(y, z)=d(x, z), can be given a plausible formulation in terms of the order on S • S; we want (xyz) to hold in this formulation if and only if it does with respect to the induced metric. More generally, a (geod- esic) segment can be defined in terms of the relation (xyz), and we want such a set to be a geodesic segment in the usual sense relative to the induced metric. In terms of a mapping (x, y)~xy of S• S onto P, the conditions we impose are: (1) this mapping induces a non-discrete Hausdorff uniformity on S; (2) the mapping is continuous in each variable to the order topology of P; (3) S is complete; (4) any point of S can be reached from any other point in finitely many small steps; (5) a strong form of M-convexity holds. Precise formulation of these conditions is given in w 1 and w below. Under these con- ditions, a metric d which is canonical in the above sense exists and is essentially unique. Moreover, S is arc-wise connected, and (xyz) holds if and only if y lies on a curve from x to z having length d(x, z). Applications of these results to the characterization of G-spaces in the sense of BUSEMANN [3] by order relations are given in w Several authors have investigated the connections between mappings from S x S to an ordered set and uniformities on S, e.g. APPERT [1] and the references there, KALISCH [5], and COHEN and GOFFMAN [4]; the question of introducing geometric entities, such as geodesic segments, in terms of orderings is not considered, however. The present work is more in the spirit of the investiga- tions of the foundations of geometry, e.g. PIERI'S axioms for Euclidean ge- ometry (see [2]), which use only the concept "y is equidistant from x and z". * The preparation of this paper was supported in part by the Air Force Office of Scientific Research Grant AF-AFOSR-736-65 and the National Science Foundation Grant NSF GP 5628.