ON THE ALEKSANDROV-RASSIAS PROBLEM AND THE GEOMETRIC INVARIANCE IN HILBERT SPACES

Abstract

Let X and Y be normed real vector spaces. A mapping f:X-> Y is called preserving the distance r if for all x, y of X with |x-yk_X=r then |f(x)-f(y)|=r. In this paper, we provide an overall account of the development of the Aleksandrov-Rassias problem in Hilbert spaces. Let X and Y be real Hilbert spaces with dimX geq 22. f : X->Y satisfies that if f preserves some kind of triangles invariant, that is, for arbitrary three points p1, p2 and p3 forming a triangle with the side length a, b and c, where a, b are positive constants, f(p1),  f(p2) and f(p3) also form a rectangular triangle the same size as p1p2p3,  then f is a linear isometry up to translation, and f preserves all geometric figures invariant. And if f preserves a kind of parallelograms invariant, then f is also a linear isometry up to translation and f preserves all geometric figures invariant.