A metric space for which every two points can be joined by a geodesic segment is called a geodesic space.
A metric space ((X,d)) is called a geodesic space if every two points of X are joined by a geodesic segment.
Let D ∈ ( 0, ∞ ], then ( X, d ) is called a D-geodesic space if any two points of X with their distance smaller than D are joined by a geodesic segment.
The space ( X, d ) is said to be a geodesic space if any two points of X are joined by a geodesic segment, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for any x, y ∈ X, which is denoted by [ x, y ] and is called the segment joining x and y.
For D ∈ ( 0, + ∞ ], the space X is called a D-geodesic space if every two points of X with their distance smaller than D are joined by a geodesic segment. An ∞-geodesic space is simply called a geodesic space. The space X is said to be uniquely geodesic (D-uniquely geodesic) if there is exactly one geodesic segment joining x and y for each x, y ∈ X (for x, y ∈ X with ρ ( x, y ) < D ).
A metric space X is a geodesic space (r-geodesic space) if every two points of X (every two points with distance smaller than r) are joined by a geodesic segment, and X is a uniquely geodesic space (r-uniquely geodesic space) if there is exactly one geodesic segment joining x and y for any (x, yin X) (for any (x, yin X) with (d x,y)< r)).
A geodesic segment is denoted by [ x, y ], if it is unique.
The image of c is called a geodesic segment joining x and y, which is denoted by [ x, y ], whenever such a segment exists uniquely.
For (x, y in X), the image of a geodesic c with endpoints x, y is called a geodesic segment joining x and y, and is denoted by ([x, y]).
The image of ξ is called a geodesic segment joining x and y which when unique is denoted by ([x,y]).
The image of c is called a geodesic segment joining x and y which when unique is denoted by seg[x, y].
