Planes for which the lines are the shortest paths between points
Ralph Alexander · Illinois Journal of Mathematics · 1978
This paper treats the planar case of Hilbert's fourth problem [14] by a direct geometric argument which is entirely free from the notion of differentiability.The key idea (Lemma 1) is a combinatorial version of Crofton's arclength formula which can be established strictly on the basis of Hilbert's simple axioms of plane incidence and order.In principle this allows a purely axiomatic treatment of the problem, although we will use the more convenient language of convexity, general topology and measure theory.Even though the removal of cumbersome variational techniques is a major benefit, there are other virtues of the present approach.Since there is no need to assume a Euclidean incidence structure, the theorem of Desargues may be ignored.This greatly increases the method's scope.Also, the three basic lemmas on the foundations of integral geometry require absolutely no continuity assumptions.This ultimately allows the treatment of discontinuous path-length functions and a clear description of the possible sets of discontinuities.Briefly stated, the fourth problem requests an investigation of the various geometries obtained by replacing the usual triangle congruence axiom with the requirement that the triangle inequality hold for the sides of any triangle.The axiom of parallels is dropped, but the remainder of the Hilbert axiom scheme is retained.The heart of the problem is to identify all metrics on such a plane which add continuously along the lines.They give rise to the continuous path- length functions for which "the lines are the shortest connection between points".In order to avoid an axiomatic description we make the following four assumptions about the planes to be investigated: (1) the points of the plane carry a topology making the plane homeomorphic to the Euclidean plane; (2) the lines are certain pointsets which are homeomorphic to Euclidean lines in the relative topology; (3) two distinct points lie on precisely one line; (4) each line separates the plane into two distinct nonempty open convex sets, or equiv- alently, the axiom of Pasch is valid.The method does extend to some other general line-systems, certainly the projective plane, but we will use the above conditions which allow a simple exposition.