Let X and Y be two real normed vector spaces. A mapping /: X —► Y preserves unit distance in both directions iff for all x, y e X with ||jc — y|| = l it follows that ||/(jc) /0>)|| = 1 and conversely. In this paper we shall study, instead of isometries, mappings satisfying the weaker assumption that they preserve unit distance in both directions. We shall prove that such mappings are not very f...

A Banach space $X$ has the $Mazur$-$Ulam$ $property$ if any isometry from unit sphere of onto other $Y$ extends to a linear spaces $X,Y$. is called $smooth$ ball unique supporting functional at each point sphere. We prove that non-smooth 2-dimensional Mazur-Ulam property.