Criteria for unique metric lines (Q1068380)

A class of complete, metrically convex, externally convex metric spaces M (strictly larger than the class of Banach spaces) is described such that the following properties are equivalent in each M: (i) two different points lie in a unique metric line; (ii) the foot of a point on a metric line is unique; (iii) the distance of a point p to points q of a metric line strictly increases as q moves along the line in a direction from a foot of p; (iv) the spheres of M are strictly convex.