Ein verschärfter und verallgemeinerter Satz von Alexandrov

Abstract

In this paper we prove the following theorem: Suppose that n≥3 and 1≤j<n. Let d be the following generalized pseudo-euclidean distance function on ℝⁿ

$$(\forall a,b) d(a,b) : = \sum\limits_{\nu = 1}^j { (a_\nu - b_\nu )^2 - \sum\limits_{\nu = j + 1}^n { (a_\nu - b_\nu )^2 .} }$$

If a function f:ℝⁿ→ℝⁿ satisfies the condition:

$$(\forall x,y \in \mathbb{R}^n ) d(f(x),f(y)) = 0 \Leftrightarrow d(x,y) = 0,$$

((*))

then f is affine. Moreover, f preserves distances up to a constant factor C≠0, i.e. d(f(x),f(y))=C·d(x,y) for every x,y.

In contrast to Alexandrov's result [1] we do not assume that f is bijective, and we also do not assume that j=n−1.

A very important part of our proof will be the discussion of a functional equation.