A finite-dimensional Teichmüller space is a geodesic straight space in the sense of Busemann. However, any infinite-dimensional Teichmüller space is not a geodesic straight space. At some kind of points in infinite-dimensional Teichmüller spaces, the geometric behavior of these points with respect to the base point is very “bad”. Such phenomena were sufficiently investigated. The study led to a complete characterization of such points and promoted a better understanding of the geometry of Teichmüller spaces. This topic is closely connected with the extremal problem of quasi-conformal mappings.