ABSTRACT
This chapter introduces the minimal generalized galleries in the setting of Coxeter complexes. The existence of minimal generalized galleries is proven by means of the geometrical realization of a Coxeter Complex as a decomposition of an euclidean space by simplicial cones. A unicity result about generalized galleries with associated minimal galleries of types, i.e. galleries of types defined by minimal generalized galleries, is proven. An expression of the convex hull of two facets is obtained. It allows one to prove that a Minimal Generalized Gallery is contained in the convex hull of its extremities. The chapter describes the type of relative position associated to Minimal Generalized Galleries of types. The correspondence between Minimal Generalized Galleries issued from a chamber C and a set of words in is extended to a correspondence between the generalized galleries issued from C and words in a set larger than.
