A contribution to a theorem of Ulam and Mazur

Unit-sphere preserving mappings

“…S. Beckman and D. A. Quarles [2] solved the Aleksandrov problem for finite-dimensional real Euclidean spaces X = R n (see also [3,4,5,6,7,8,10,11,12,13,14,16,17,18,19]): Theorem 1.1 (Theorem of Beckman and Quarles). If a mapping f : R n → R n (2 ≤ n < ∞) preserves a distance r > 0, then f is a linear isometry up to translation.…”

An inequality for distances between 2n points and the Aleksandrov–Rassias problem

“…Such a problem is called the Aleksandrov- Rassias problem. For strictly convex vector spaces, an answer was given by W. Benz [3] (see also [4]): In what follows, we will label the vertices of any (possibly degenerate) parallelogram by p 11 , p 12 , p 21 , and p 22 as we see in the left-hand side of Fig. 1.…”

A contribution to the Aleksandrov conservative distance problem in two dimensions

“…As far as we know, the original version of Aleksandrov problem was solved only for a few concrete normed spaces (see [22] concerning p-norms, and [13] where the norm is not strictly convex), all of them are two-dimensional. Some general results are known for modified versions, for instance in [5] W. Benz and H. Berens investigated the case when the transformation preservers distance 1 and n for some n ∈ N, n > 1. We also mention the paper [18] of T. M. Rassias and P.Šemrl where they assumed that φ is onto and it preserves distance 1 in both directions.…”