In metric geometry, a geodesic bicombing distinguishes a class of geodesics of a metric space. The study of metric spaces with distinguished geodesics traces back to the work of the mathematician Herbert Busemann. The convention to call a collection of paths of a metric space bicombing is due to William Thurston. By imposing a weak global non-positive curvature condition on a geodesic bicombing several results from the theory of CAT(0) spaces and Banach space theory may be recovered in a more general setting.
Definition Let ( X , d ) {\displaystyle (X,d)} be a metric space. A map σ : X × X × [ 0 , 1 ] → X {\displaystyle \sigma \colon X\times X\times [0,1]\to X} is a geodesic bicombing if for all points x , y ∈ X {\displaystyle x,y\in X} the map σ x y ( ⋅ ) := σ ( x , y , ⋅ ) {\displaystyle \sigma _{xy}(\cdot ):=\sigma (x,y,\cdot )} is a unit speed metric geodesic from x {\displaystyle x} to y {\displaystyle y} , that is, σ x y ( 0 ) = x {\displaystyle \sigma _{xy}(0)=x} , σ x y ( 1 ) = y {\displaystyle \sigma _{xy}(1)=y} and d ( σ x y ( s ) , σ x y ( t ) ) = | s − t | d ( x , y ) {\displaystyle d(\sigma _{xy}(s),\sigma _{xy}(t))=\vert s-t\vert d(x,y)} for all real numbers s , t ∈ [ 0 , 1 ] {\displaystyle s,t\in [0,1]} .
Different classes of geodesic bicombings A geodesic bicombing σ : X × X × [ 0 , 1 ] → X {\displaystyle \sigma \colon X\times X\times [0,1]\to X} is:
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