Metric matroids and metric buildings
Alexandre V Borovik · MIMS EPrints (University of Southampton) · 2008
These notes represent fragments of an old joint work with I. M. Gelfand; I put in them some material which did not find its way in our book with Neil White "Coxeter Matroids", but, I believe, deserves to be documented. In particular, I use the old term "W-matroid" for what later became a "Coxeter matroid for Coxeter group W". We discuss realizations of W-matroids by retractions of buildings and their convexity properties. We define, in terms of metric properties of retractions, some class of geometric objects which includes,as partial cases, buildings and R-trees. For some of these new geometric structures `abstract' convexity properties of retractions have a natural interpretation in terms of usual Euclidean convexity.