COMBINATORIAL VALUATIONS IN THE SPACE OF PREGEODESICS

摘要

This is the second attempt to present the combinatorial theory of pregeodesics, connecting it with Hilbert's Fourth Problem on 2-manifolds. The first attempt was done in 1996 in the author's article . The years that elapsed witnessed considerable progress in the theory of combinatorial valuations in the classical integral geometry spaces, in particular in the spaces of planes in IR~3 and of lines in IR~3. We especially mention the result that identifies the sufficiently smooth translation invariant (but otherwise general) valuations in the latter two spaces as combinatorial valuations . The success of valuations in the theory of pregeodesics was also essential: along with the work directed to reach a higher degree of precision and rigor as well as deeper understanding, usually necessary on the first stages of development of a novel subject, new important facts have also been added. The present article is aimed at presenting the advances in the latter field both in methodology and in the search of new facts. Thus, Theorem 4 below identifies all continuous valuations in the spaces of pregeodesics as combinatorial valuations. In another instance, Theorem 10 below points at a pointwise convexity condition that was not anticipated in, which together with the Stable additivity condition solves the Hilbert's Fourth Problem for pregeodesics within the classical environment. As for the methodological improvements, we mention that a universal proof of the main decomposition theorem together with the algorithm for its combinatorial coefficients can now be derived from a general theorem on demarcation of open Moebius band by special circles (i.e. topological circles that do not split the Moebius band in two components). The author plans to soon publish the proof of that theorem from Moebius band geometry elsewhere. We note that the book contained four or five different ad hoc proofs of the main decomposition theorem in different concrete spaces (the general concept of pregeodesics remained beyond the scope of ). This development suggests a research direction for other integral geometry spaces.