The Beckman - Quarles theorem for continuous mappings from R n to C

Let φ : Cn×Cn → C, φ((x1, ..., xn), (y1, ..., yn)) = (x1−y1)+ ...+ (xn − yn). We say that f : Rn → Cn preserves distance d ≥ 0 if for each x, y ∈ Rn φ(x, y) = d2 implies φ(f(x), f(y)) = d2. We prove that if x, y ∈ Rn (n ≥ 3) and |x − y| = ( √ 2 + 2/n)k · (2/n)l (k, l are nonnegative integers) then there exists a finite set {x, y} ⊆ Sxy ⊆ Rn such that each unit-distance preserving mapping from Sxy to C n preserves the distance between x and y. It implies that each continuous map from Rn to Cn (n ≥ 3) preserving unit distance preserves all distances. The classical Beckman-Quarles theorem states that each unit-distance preserving mapping from R to R (n ≥ 2) is an isometry, see [1], [2] and [7]. Author’s discrete form of this theorem ([8],[9]) states that if x, y ∈ R (n ≥ 2) and |x−y| is an algebraic number then there exists a finite set {x, y} ⊆ Sxy ⊆ R such that each unit-distance preserving mapping from Sxy to R n preserves the distance between x and y. Let φ : C ×Cn → C, φ((x1, ..., xn), (y1, ..., yn)) = (x1 − y1) + ...+ (xn − yn) . We say that f : R → C preserves distance d ≥ 0 if for each x, y ∈ R φ(x, y) = d implies φ(f(x), f(y)) = d. Our goal is to prove that each continuous map from R to C (n ≥ 3) preserving unit distance preserves all distances. It requires some technical propositions. Proposition 1 (cf. [3], [4]). For each points c1, ..., cn+2 ∈ C their Cayley-Menger determinant equals 0 i.e. det   0 1 1 ... 1 1 φ(c1, c1) φ(c1, c2) ... φ(c1, cn+2) 1 φ(c2, c1) φ(c2, c2) ... φ(c2, cn+2) ... ... ... ... ... 1 φ(cn+2, c1) φ(cn+2, c2) ... φ(cn+2, cn+2)   = 0.