METRIC SPACES WITH UNIQUE PRETANGENT SPACES. CONDITIONS OF THE UNIQUENESS

We find necessary and sufficient conditions for an arbitrary metric space X to have a unique pretangent space at a marked point a is an element of X. Applying this general result we show that each logarithmic spiral has a unique pretangent space at the asymptotic point. Unbounded multiplicative subgroups of C* = C\{0} having unique pretangent spaces at zero are characterized as lying either on the positive real semiaxis or on logarithmic spirals. Our general uniqueness conditions in the case X subset of R make it also possible to characterize the points of the ternary Cantor set having unique Pretangent spaces.