A satisfactory theory of 2-norms and n-norms on a linear space has been introduced and developed by Gähler 12. Following Misiak 3, Kim and Cho 4, and Malčeski 5 developed the theory of n-normed space. In 6, Gunawan and Mashadi gave a simple way to derive an (n - 1)-norm from the n-norms and realized that any n-normed space is an (n - 1)-normed space. Different authors introduced the definitions of fuzzy norms on a linear space. Cheng and Mordeson 7 and Bag and Samanta 8 introduced a concept of fuzzy norm on a linear space. The concept of fuzzy n-normed linear spaces has been studied by many authors (see 49).

Recently, Somasundaram and Beaula 10 introduced the concept of 2-fuzzy 2-normed linear space or fuzzy 2-normed linear space of the set of all fuzzy sets of a set. The authors gave the notion of α-2-norm on a linear space corresponding to the 2-fuzzy 2-norm by using some ideas of Bag and Samanta 8 and also gave some fundamental properties of this space.

In 1932, Mazur and Ulam 11 proved the following theorem.

Mazur-Ulam Theorem. Every isometry of a real normed linear space onto a real normed linear space is a linear mapping up to translation.

Baker 12 showed an isometry from a real normed linear space into a strictly convex real normed linear space is affine. Also, Jian 13 investigated the generalizations of the Mazur-Ulam theorem in F*-spaces. Rassias and Wagner 14 described all volume preserving mappings from a real finite dimensional vector space into itself and Väisälä 15 gave a short and simple proof of the Mazur-Ulam theorem. Chu 16 proved that the Mazur-Ulam theorem holds when X is a linear 2-normed space. Chu et al. 17 generalized the Mazur-Ulam theorem when X is a linear n-normed space, that is, the Mazur-Ulam theorem holds, when the n-isometry mapped to a linear n-normed space is affine. They also obtain extensions of Rassias and Šemrl's theorem 18. Moslehian and Sadeghi 19 investigated the Mazur-Ulam theorem in non-archimedean spaces. Choy et al. 20 proved the Mazur-Ulam theorem for the interior preserving mappings in linear 2-normed spaces. They also proved the theorem on non-Archimedean 2-normed spaces over a linear ordered non-Archimedean field without the strict convexity assumption. Choy and Ku 21 proved that the barycenter of triangle carries the barycenter of corresponding triangle. They showed the Mazur-Ulam problem on non-Archimedean 2-normed spaces using the above statement. Xiaoyun and Meimei 22 introduced the concept of weak n-isometry and then they got under some conditions, a weak n-isometry is also an n-isometry. Cobzaş 23 gave some results of the Mazur-Ulam theorem for the probabilistic normed spaces as defined by Alsina et al. 24. Cho et al. 25 investigated the Mazur-Ulam theorem on probabilistic 2-normed spaces. Alaca 26 introduced the concepts of 2-isometry, collinearity, 2-Lipschitz mapping in 2-fuzzy 2-normed linear spaces. Also, he gave a new generalization of the Mazur-Ulam theorem when X is a 2-fuzzy 2-normed linear space or ℑ(X) is a fuzzy 2-normed linear space. Kang et al. 27 proved that the Mazur-Ulam theorem holds under some conditions in non-Archimedean fuzzy normed space. Kubzdela 28 gave some new results for isometries, Mazur-Ulam theorem and Aleksandrov problem in the framework of non-Archimedean normed spaces. The Mazur-Ulam theorem has been extensively studied by many authors (see 2930).

In the present article, we introduce the concept of 2-fuzzy n-normed linear space or fuzzy n-normed linear space of the set of all fuzzy sets of a non-empty set. We define the concepts of n-isometry, n-collinearity, n-Lipschitz mapping in this space. Also, we generalize the Mazur-Ulam theorem, that is, when X is a 2-fuzzy n-normed linear space or ℑ(X) is a fuzzy n-normed linear space, the Mazur-Ulam theorem holds. It is moreover shown that each n-isometry in 2-fuzzy n-normed linear spaces is affine.

Definition 2.1(31) Let n ∈ ℕ and let X be a real vector space of dimension d ≥ n. (Here we allow d to be infinite.) A real-valued function ∥●, ..., ●∥ on X × ⋯ × X ︸ n satisfying the following properties

(1) ∥x₁, x₂, ..., x_n∥ = 0 if and only if x₁, x₂, ..., x_n are linearly dependent,

(2) ∥x₁, x₂, ..., x_n∥ is invariant under any permutation,

(3) ∥x₁, x₂, ..., αx_n∥ = |α| ∥x₁, x₂, ..., x_n∥ for any α ∈ ℝ,

(4) ∥x₁, x₂, ..., x_n-1, y + z∥ ≤ ∥x₁, x₂, ..., x_n-1, y∥ + ∥x₁, x₂, ..., x_n-1, z∥, is called an n-norm on X and the pair (X, ∥●, ..., ●∥) is called an n-normed linear space.

Definition 2.2 9 Let X be a linear space over S (field of real or complex numbers). A fuzzy subset N of Xⁿ × ℝ (ℝ, the set of real numbers) is called a fuzzy n-norm on X if and only if:

(N1) For all t ∈ ℝ with t ≤ 0, N(x₁, x₂, ..., x_n, t) = 0,

(N2) For all t ∈ ℝ with t > 0, N(x₁, x₂, ..., x_n, t) = 1 if and only if x₁, x₂, ..., x_n are linearly dependent,

(N3) N(x₁, x₂, ..., x_n, t) is invariant under any permutation of x₁, x₂, ..., x_n,

(N4) For all t ∈ ℝ with t > 0, N ( x 1 , x 2 , … , λ x n , t ) = N x 1 , x 2 , … , x n , t λ , if λ ≠ 0, λ ∈ S,

(N5) For all s, t ∈ ℝ

N ( x 1 , x 2 , … , x n + x n ′ , s + t ) ≥ min { N ( x 1 , x 2 , … , x n , s ) , N ( x 1 , x 2 , … , x n ′ , t ) } ,

(N6) N(x₁, x₂, ..., x_n, t) is a non-decreasing function of t ∈ ℝ and lim t → ∞ N ( x 1 , x 2 , … , x n , t ) = 1 .

Then (X, N) is called a fuzzy n-normed linear space or in short f-n-NLS.

Theorem 2.1 9 Let (X, N) be an f-n-NLS. Assume that

(N7) N(x₁, x₂, ..., x_n,t) > 0 for all t > 0 implies that x₁, x₂, ..., x_n are linearly dependent.

Define

| | x 1 , x 2 , … , x n | | α = inf { t : N ( x 1 , x 2 , … x n , t ) ≥ α , α ∈ ( 0 , 1 ) } .

Then {∥●, ●, ..., ●∥_α : α ∈ (0, 1)} is an ascending family of n-norms on X.

We call these n-norms as α-n-norms on X corresponding to the fuzzy n-norm on X.

Definition 2.3 Let X be any non-empty set and ℑ(X) the set of all fuzzy sets on X. For U, V ∈ ℑ(X) and λ ∈ S the field of real numbers, define

U + V = { ( x + y , ν ∧ μ ) : ( x , ν ) ∈ U , ( y , μ ) ∈ V }

and λU = {(λx, ν): (x, ν) ∈ U}.

Definition 2.4 A fuzzy linear space X ^ = X × ( 0 , 1 ] over the number field S, where the addition and scalar multiplication operation on X are defined by (x, ν) + (y, μ) = (x + y, ν∧μ), λ(x, ν) = (λx, ν) is a fuzzy normed space if to every ( x , ν ) ∈ X ^ there is associated a non-negative real number, ∥(x, ν)∥, called the fuzzy norm of (x, ν), in such away that

(i) ∥(x, ν)∥ = 0 iff x = 0 the zero element of X, ν ∈ (0, 1],

(ii) ∥λ(x, ν)∥ = |λ| ∥(x, ν)∥ for all (x, ν) ∈X ^ and all λ ∈ S,

(iii) ∥(x, ν) + (y, μ) || ≤ ∥(x, ν ∧ μ)∥ + ∥(y, ν ∧ μ)∥ for all ( x , ν ) , ( y , μ ) ∈ X ^ ,

(iv) ∥(x, ∨_tν_t)∥ = ∧_t ∥(x, ν_t)∥ for all ν_t ∈ (0, 1].

In this section, we define the concepts of 2-fuzzy n-normed linear spaces and α-n-norms on the set of all fuzzy sets of a non-empty set.

Definition 3.1 Let X be a non-empty and ℑ(X) be the set of all fuzzy sets in X. If f ∈ ℑ(X) then f = {(x, μ): x ∈ X and μ ∈ (0, 1]}. Clearly f is bounded function for |f(x)| ≤ 1. Let S be the space of real numbers, then ℑ(X) is a linear space over the field S where the addition and scalar multiplication are defined by

f + g = { ( x , μ ) + ( y , η ) } = { ( x + y , μ ∧ η ) : ( x , μ ) ∈ f and ( y , η ) ∈ g }

and

λ f = { ( λ x , μ ) : ( x , μ ) ∈ f }

where λ ∈ S.

The linear space ℑ(X) is said to be normed linear space if, for every f ∈ ℑ(X), there exists an associated non-negative real number ∥f∥ (called the norm of f) which satisfies

(i) ∥f∥ = 0 if and only if f = 0. For

| | f | | = 0 ⇔ { | | ( x , μ ) | | : ( x , μ ) ∈ f } = 0 ⇔ x = 0 , μ ∈ ( 0 , 1 ] ⇔ f = 0 .

(ii) ∥λf∥ = |λ| ∥f∥, λ ∈ S. For

| | λ f | | = { | | λ ( x , μ ) | | : ( x , μ ) ∈ f , λ ∈ S } = { | λ | | | ( x , μ ) | | : ( x , μ ) ∈ f } = | λ | | | f | | .

(iii) ∥f + g∥ ≤ ∥f∥ + ∥g∥ for every f, g ∈ ℑ(X). For

| | f + g | | = { | | ( x , μ ) + ( y , η ) | | : x , y ∈ X , μ , η ∈ ( 0 , 1 ] } = { | | ( x + y ) , ( μ ∧ η ) | | : x , y ∈ X , μ , η ∈ ( 0 , 1 ] } = { | | ( x , μ ∧ η ) | | + | | ( y , μ ∧ η ) | | : ( x , μ ) ∈ f , ( y , η ) ∈ g } = | | f | | + | | g | | .

Then (ℑ(X),∥●∥) is a normed linear space.

Definition 3.2 A 2-fuzzy set on X is a fuzzy set on ℑ(X).

Definition 3.3 Let X be a real vector space of dimension d ≥ n (n ∈ ℕ) and ℑ(X) be the set of all fuzzy sets in X. Here we allow d to be infinite. Assume that a [0, 1]-valued function ∥●, ..., ●∥ on ℑ ( X ) × ⋯ × ℑ ( X ) ︸ n satisfies the following properties

(1) ∥f₁, f₂, ..., f_n∥ = 0 if and only if f₁, f₂, ..., f_n are linearly dependent,

(2) ∥f₁, f₂, ..., f_n∥ is invariant under any permutation,

(3) ∥f₁, f₂, ..., λf_n∥ = |λ| ∥f₁, f₂, ..., f_n∥ for any λ ∈ S,

(4) ∥f₁, f₂, ..., f_n-1, y + z∥ ≤ ∥f₁, f₂, ..., f_n-1, y∥ + ∥f₁, f₂, ..., f_n-1, z∥.

Then (ℑ(X),∥●,...,●∥) is an n-normed linear space or (X, ∥●, ..., ●∥) is a 2-n-normed linear space.

Definition 3.4 Let ℑ(X) be a linear space over the real field S. A fuzzy subset N of ℑ ( X ) × ⋯ × ℑ ( X ) ︸ n × ℝ is called a 2-fuzzy n-norm on X (or fuzzy n-norm on ℑ(X)) if and only if

(2-N1) for all t ∈ ℝ with t ≤ 0, N(f₁, f₂, ..., f_n, t) = 0,

(2-N2) for all t ∈ ℝ with t > 0, N(f₁, f₂, ..., f_n, t) = 1 if and only if f₁, f₂, ..., f_n are linearly dependent,

(2-N3) N(f₁, f₂, ..., f_n, t) is invariant under any permutation of f₁, f₂, ..., f_n,

(2-N4) for all t ∈ ℝ with t > 0, N(f₁, f₂, ..., λf_n, t) = N(f₁, f₂, ..., f_n, t/|λ|), if λ ≠ 0, λ ∈ S,

(2-N5) for all s, t ∈ ℝ,

N ( f 1 , f 2 , … , f n + f n ′ , s + t ) ≥ min { N ( f 1 , f 2 , … , f n , s ) , N ( f 1 , f 2 , … , f n ′ , t ) } ,

(2-N6) N(f₁, f₂, ..., f_n, ·): (0, ∞) → [0, 1] is continuous,

(2-N7) lim t → ∞ N ( f 1 , f 2 , … , f n , t ) = 1 .

Then ℑ(X), N) is a fuzzy n-normed linear space or (X, N) is a 2-fuzzy n-normed linear space.

Remark 3.1 In a 2-fuzzy n-normed linear space (X, N), N(f₁, f₂, ..., f_n, ·) is a non-decreasing function of ℝ for all f₁, f₂,...,f_n ∈ ℑ(X).

Remark 3.2 From (2-N4) and (2-N5), it follows that in a 2-fuzzy n-normed linear space,

(2-N4) for all t ∈ ℝ with t > 0, N ( f 1 , f 2 , … , λ f i , … , f n , t ) = N f 1 , f 2 , … , f i , … , f n , t | λ | , if λ ≠ 0, λ ∈ S,

(2-N5) for all s, t ∈ ℝ,

N ( f 1 , f 2 , … , f i + f ′ i , … , f n , s + t ) ≥ min { N ( f 1 , f 2 , … , f i , … , f n , s ) , N ( f 1 , f 2 , … , f ′ i , … , f n , t ) } .

The following example agrees with our notion of 2-fuzzy n-normed linear space.

Example 3.1 Let (ℑ(X),∥●,●,...,●∥) be an n-normed linear space as in Definition 3.3. Define

N ( f 1 , f 2 , … , f n , t ) = t t + | | f 1 , f 2 , … , f n | | i f t > 0 , t ∈ ℝ , 0 i f t ≤ 0

for all ( f 1 , f 2 , … , f n ) ∈ ℑ ( X ) × ⋯ × ℑ ( X ) ︸ n . Then (X, N) is a 2-fuzzy n-normed linear space.

Solution. (2-N1) For all t ∈ ℝ with t ≤ 0, by definition, we have N(f₁, f₂, ..., f_n, t) = 0.

(2-N2) For all t ∈ ℝ with t > 0,

N ( f 1 , f 2 , … , f n , t ) = 1 ⇔ t t + | | f 1 , f 2 , … , f n | | = 1 ⇔ t = t + | | f 1 , f 2 , … , f n | | ⇔ | | f 1 , f 2 , … , f n | | = 0 ⇔ f 1 , f 2 , … , f n  are linearly dependent .

(2-N3) For all t ∈ ℝ with t > 0,

N ( f 1 , f 2 , … , f n , t ) = t t + | | f 1 , f 2 , … , f n | | = t t + | | f 1 , f 2 , … , f n , f n - 1 | | = N ( f 1 , f 2 , … , f n , f n - 1 , t ) = ⋯ .

(2-N4) For all t ∈ ℝ with t > 0 and λ ∈ F, λ ≠ 0,

N ( f 1 , f 2 , … , f n , t / | λ | ) = t / | λ | t / | λ | + | | f 1 , f 2 , … , f n | |                   = t / | λ | ( t + | λ |   | | f 1 , f 2 , … , f n | | ) / | λ |                   = t t + | λ |   | | f 1 , f 2 , … , f n | |                   = t ( t + | | f 1 , f 2 , … , λ f n | | = N ( f 1 , f 2 , … , λ f n , t ) .

(2-N5) We have to prove

N ( f 1 , f 2 , … , f n + f n ′ , s + t ) ≥ min { f ( x 1 , f 2 , … , f n , s ) , N ( f 1 , f 2 , … , f n ′ , t ) } .

(i) s + t < 0,

(ii) s = t = 0,

(iii) s + t > 0; s > 0, t < 0; s < 0, t > 0, then the above relation is obvious. If

(iv) s > 0, t > 0, s + t > 0, then

N ( f 1 , f 2 , … , f n + f ′ n , s + t = s + t s + t + | | f 1 , f 2 , … , f n + f ′ n | | .

If

s s + | | f 1 , f 2 , … , f n | | ≥ t t + | | f 1 , f 2 , … , f ′ n | | ⇒ | | f 1 , f 2 , … , f n | | s ≤ | | x 1 , x 2 , … , x ′ n | | t ⇒ | | f 1 , f 2 , … , f n | | s + | | f 1 , f 2 , … , f ′ n | | s ≤ | | f 1 , f 2 , … , f ′ n | | t + | | f 1 , f 2 , … , f ′ n | | s ⇒ | | f 1 , f 2 , … , f n + f ′ n | | s ≤ s + t s ⋅ t | | f 1 , f 2 , … , f ′ n | | ⇒ | | f 1 , f 2 , … , f n + f ′ n | | s + t ≤ | | f 1 , f 2 , … , f ′ n | | t ⇒ s + t + | | f 1 , f 2 , … , f n + f ′ n | | s + t ≤ t + | | f 1 , f 2 , … , f ′ n | | t ⇒ s + t s + t + | | f 1 , f 2 , … , f n + f ′ n | | ≥ t t + | | f 1 , f 2 , … , f ′ n | | ⇒ N ( f 1 , f 2 , … , f n + f ′ n , s + t ) ≥ N ( f 1 , f 2 , … , f ′ n , t ) .

Similarly, if

t t + | | f 1 , f 2 , … , f ′ n | | ≥ s s + | | f 1 , f 2 , … , f n | | ⇒ N ( f 1 , f 2 , … , f n + f ′ n , s + t ) ≥ N ( f 1 , f 2 , … , f n , t ) .

Thus

N ( f 1 , f 2 , … , f n + f n ′ , s + t ) ≥ min { N ( f 1 , f 2 , … , f n , s ) , N ( f 1 , f 2 , … , f n ′ , t ) } .

(2-N6) It is clear that N(f₁, f₂, ..., f_n, ·): (0, ∞) → [0, 1] is continuous.

(2-N7) For all t ∈ ℝ with t > 0,

lim t → ∞ N ( f 1 , f 2 , … , f n , t ) = lim t → ∞ t t + | | f 1 , f 2 , … , f n | | = lim t → ∞ t t ( 1 + ( 1 / t ) | | f 1 , f 2 , … , f n | | ) = 1 ,

as desired.

As a consequence of Theorem 3.2 in 10, we introduce an interesting notion of ascending family of α-n-norms corresponding to the fuzzy n-norms in the following theorem.

Theorem 3.1 Let (ℑ(X), N) is a fuzzy n-normed linear space. Assume that

(2-N8) N(f₁, f₂, ..., f_n, t) > 0 for all t > 0 implies f₁, f₂, ..., f_n are linearly dependent.

Define

| | f 1 , f 2 , … , f n | | α = inf ( t : N ( f 1 , f 2 , … , f n , t ) ≥ α , α ∈ ( 0 , 1 ) } .

Then {∥●, ●, ..., ●∥_α : α ∈ (0, 1)} is an ascending family of n-norms on ℑ(X).

These n-norms are called α-n-norms on ℑ(X) corresponding to the 2-fuzzy n-norm on X.

Proof. (i) Let ∥f₁, ..., f_n∥_α = 0. This implies that inf {t : N(f₁, ..., f_n, t) ≥ α}. Then, N(f₁, f₂, ..., f_n, t) ≥ α > 0, for all t > 0, α ∈ (0, 1), which implies that f₁, f₂, ..., f_n are linearly dependent, by (2-N8).

Conversely, assume f₁, f₂, ..., f_n are linearly dependent. This implies that N(f₁, f₂, ..., f_n, t) = 1 for all t > 0. For all α ∈ (0, 1), inf {t : N(f₁, f₂, ..., f_n, t) ≥ α}, which implies that ∥f₁, f₂, ..., f_n∥_α = 0.

(ii) Since N(f₁, f₂, ..., f_n, t) is invariant under any permutation, ∥f₁, f₂, ..., f_n∥_α = 0 under any permutation.

(iii) If λ ≠ 0, then

| | f 1 , f 2 , … , λ f n | | α = inf { s : N ( f 1 , f 2 , … , f n , s ) ≥ α }                   = inf { s : N ( f 1 , f 2 , … , f n , s | λ | ≥ α } .

Let t = s | λ | , then

| | f 1 , f 2 , … , λ f n | | α = inf { | λ | t : N ( f 1 , f 2 , … , f n , t ) ≥ α } = | λ | inf { t : N ( f 1 , f 2 , … , f n , t ) ≥ α } = | λ | | | f 1 , f 2 , … , f n | | α .

If λ = 0, then

| | f 1 , f 2 , … , λ f n | | α = | | f 1 , f 2 , … , 0 | | α = 0 = 0 | | f 1 , f 2 , … , f n | | α                   = | λ |   | | f 1 , f 2 , … , f n | | α , ∀ λ ∈ S ( field ) .

(iv)

f 1 , f 2 , … , f n α + f 1 , f 2 , … , f ′ n α = inf { t : N ( f 1 , f 2 , … , f n , t ) ≥ α } + inf ( s : N ( f 1 , f 2 , … , f ′ n , s ) ≥ α } = inf { t + s : N ( f 1 , f 2 , … , f n , t ) ≥ α , N ( f 1 , f 2 , … , f ′ n , s ) ≥ α } ≥ inf { t + s : N ( f 1 , f 2 , … , f n + f ′ n , t + s ) ≥ α } , ≥ inf { r : N ( f 1 , f 2 , … , f n + f ′ n , r ) ≥ α } , r = t + s = f 1 , f 2 , … , f n + f ′ n α .

Hence

| | f 1 , f 2 , … , f n + f n ′ | | α ≤ | | f 1 , f 2 , … , f n | | α + | | f 1 , f 2 , … , f n ′ | | α .

Thus {∥●, ●, ..., ●∥_α : α ∈ (0, 1)} is an α-n-norm on X.

Let 0 < α₁ < α₂. Then,

f 1 , f 2 , … , f n α 1 = inf { t : N ( f 1 , f 2 , … , f n , t ) ≥ α 1 } , f 1 , f 2 , … , f n α 2 = inf { t : N ( f 1 , f 2 , … , f n , t ) ≥ α 2 } .

As α₁ < α₂,

{ t : N ( f 1 , f 2 , … , f n , t ) ≥ α 2 } ⊂ { t : N ( f 1 , f 2 , … , f n , t ) ≥ α 1 }

implies that

inf { t : N ( f 1 , f 2 , … , f n , t ) ≥ α 2 } ≥ inf { t : N ( f 1 , f 2 , … , f n , t ) ≥ α 1 }

which implies that

| | f 1 , f 2 , … , f n | | α 2 ≥ | | f 1 , f 2 , … , f n | | α 1 .

Hence {∥●, ●, ..., ●∥_α : α ∈ (0, 1)} is an ascending family of α-n-norms on x corresponding to the 2-fuzzy n-norm on X.

In this section, we give a new generalization of the Mazur-Ulam theorem when X is a 2-fuzzy n-normed linear space or ℑ(X) is a fuzzy n-normed linear space. Hereafter, we use the notion of fuzzy n-normed linear space on ℑ(X) instead of 2-fuzzy n-normed linear space on X.

Definition 4.1 Let ℑ(X) and ℑ(X) be fuzzy n-normed linear spaces and Ψ : ℑ(X) → ℑ(Y) a mapping. We call Ψ an n-isometry if

f 1 - f 0 , … , f n - f 0 α = Ψ ( f 1 ) - Ψ ( f 0 ) , … , Ψ ( f n ) - Ψ ( f 0 ) β

for all f₀, f₁, f₂,...,f_n ∈ ℑ(X) and α, β ∈ (0, 1).

For a mapping Ψ, consider the following condition which is called the n-distance one preserving property (nDOPP).

( n DOPP ) Let f 0 , f 1 , f 2 , … , f n ∈ ℑ ( X ) with f 1 - f 0 , … , f n - f 0 α = 1 .

Then ∥Ψ(f₁) - Ψ(f₀), ..., Ψ(f_n) - Ψ(f₀)∥_β = 1.

Lemma 4.1 Let f₁, f₂,...,f_n ∈ ℑ(X), α ∈ (0, 1) and ħ ∈ ℝ. Then,

| | f 1 , … , f i , … , f j , … , f n | | α = | | f 1 , … , f i , … , f j + ℏ f i , … , f n | | α

for all 1 ≤ i ≠ j ≤ n.

Proof. It is obviously true.

Lemma 4.2 For f 0 , f 0 ′ ∈ ℑ ( X ) , if f₀ and f 0 ′ are linearly dependent with some direction, that is, f 0 ′ = t f 0 for some t > 0, then

| | f 0 + f 0 ′ , f 1 , … , f n | | α = | | f 0 , f 1 , … , f n | | α + | | f 0 ′ , f 1 , … , f n | | α

for all f₁, f₂,...,f_n ∈ ℑ(X) and α ∈ (0, 1).

Proof. Let f0′ = tf0 for some t > 0. Then we have

| | f 0 + f ′ 0 , f 1 , … , f n | | α = | | f 0 + t f 0 , f 1 , … , f n | | α = ( 1 + t ) | | f 0 , f 1 , … , f n | | α = | | f 0 , f 1 , … , f n | | α + t | | f 0 , f 1 , … , f n | | α = | | f 0 , f 1 , … , f n | | α + | | f ′ 0 , f 1 , … , f n | | α

for all f₁, f₂,...,f_n ∈ ℑ(X) and α ∈ (0, 1).

Definition 4.2 The elements f₀, f₁, f₂, ..., f_n of ℑ(X) are said to be n-collinear if for every i, {f_j - f_i : 0 ≤ j ≠ i ≤ n} is linearly dependent.

Remark 4.1 The elements f₀, f₁, and f₂ are said to be 2-collinear if and only if f₂ - f₀ = r(f₁ - f₀) for some real number r.

Now we define the concept of n-Lipschitz mapping.

Definition 4.3 We call Ψ an n-Lipschitz mapping if there is a κ ≥ 0 such that

| | Ψ ( f 1 ) - Ψ ( f 0 ) , … , Ψ ( f n ) - Ψ ( f 0 ) | | β ≤ k | | f 1 - f 0 , … , f n - f 0 | | α

for all f₀, f₁, f₂,...,f_n ∈ ℑ(X) and α, β ∈ (0, 1). The smallest such κ is called the n-Lipschitz constant.

Lemma 4.3 Assume that if f₀, f₁, and f₂ are 2 -collinear then Ψ(f₀), Ψ(f₁) and Ψ(f₂) are 2-collinear, and that Ψ satisfies (nDOPP). Then Ψ preserves the n-distance k for each k ∈ ℕ.

Proof. Suppose that there exist f₀, f₁ ∈ ℑ(X) with f₀ ≠ f₁ such that Ψ(f₀) = Ψ(f₁). Since dimℑ(X) ≥ n, there are f₂,...,f_n ∈ ℑ(X) such that f₁ - f₀, f₂ - f₀, ..., f_n - f₀ are linearly independent. Since ∥f₁ - f₀, f₂ - f₀, ..., f_n - f₀∥_α ≠ 0, we can set

z 2 = f 0 + f 2 - f 0 | | f 1 - f 0 , f 2 - f 0 , … , f n - f 0 | | α .

Then we have

| | f 1 - f 0 , z 2 - f 0 , f 3 - f 0 , … , f n - f 0 | | α = f 1 - f 0 , f 2 - f 0 | | f 1 - f 0 , f 2 - f 0 , … , f n - f 0 | | α , f 3 - f 0 , … , f n - f 0 α = 1 .

Since Ψ preserves the unit n-distance,

| | Ψ ( f 1 ) - Ψ ( f 0 ) , Ψ ( z 2 ) - Ψ ( f 0 ) , … , Ψ ( f n ) - Ψ ( f 0 ) | | β = 1 .

But it follows from Ψ (f₀) = Ψ (f₁) that

| | Ψ ( f 1 ) - Ψ ( f 0 ) , Ψ ( z 2 ) - Ψ ( f 0 ) , … , Ψ ( f n ) - Ψ ( f 0 ) | | β = 0 ,

which is a contradiction. Hence, Ψ is injective.

Let f₀, f₁, f₂, ..., f_n be elements of ℑ(X), k ∈ ℕ and

| | f 1 - f 0 , f 2 - f 0 , … , f n - f 0 | | α = k .

We put

g i = f 0 + i k ( f 1 - f 0 ) , i = 0 , 1 , … , k .

Then

| | g i + 1 - g i , f 2 - f 0 , … , f n - f 0 | | α = f 0 + i + 1 k ( f 1 - f 0 ) - f 0 + i k ( f 1 - f 0 ) , f 2 - f 0 , … , f n - f 0 α = 1 k ( f 1 - f 0 ) , f 2 - f 0 , … , f n - f 0 α = i k | | f 1 - f 0 , f 2 - f 0 , … , f n - f 0 | | α = k k = 1

for all i = 0, 1, ..., k - 1. Since Ψ satisfies (nDOPP),

| | Ψ ( g i + 1 ) - Ψ ( g i ) , Ψ ( f 2 ) - Ψ ( f 0 ) … , Ψ ( f n ) - Ψ ( f 0 ) | | β = 1

for all i = 0, 1, ..., k - 1. Since g₀, g₁, and g₂ are 2-collinear, Ψ (g₀), Ψ(g₁) and Ψ(g₂) are also 2-collinear. Thus there is a real number r₀ such that Ψ(g₂) - Ψ(g₁) = r₀ (Ψ(g₁) - Ψ(g₀)). It follows from (4.1) that

| | Ψ ( g 1 ) - Ψ ( g 0 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , … , Ψ ( f n ) - Ψ ( f 0 ) | | β = | | Ψ ( g 2 ) - Ψ ( g 1 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , … , Ψ ( f n ) - Ψ ( f 0 ) | | β = | | r 0 ( Ψ ( g 1 ) - Ψ ( g 2 ) ) , Ψ ( f 2 ) - Ψ ( f 0 ) , … , Ψ ( f n ) - Ψ ( f 0 ) | | β = | r 0 | | | ( Ψ ( g 1 ) - Ψ ( g 0 ) ) , Ψ ( f 2 ) - Ψ ( f 0 ) , … , Ψ ( f n ) - Ψ ( f 0 ) | | β .

Thus, we have r₀ = 1 or -1. If r₀ = -1, Ψ(g₂) - Ψ(g₁) = -Ψ(g₁) + Ψ(g₀), that is, Ψ(g₂) = Ψ(g₀). Since Ψ is injective, g₂ = g₀, which is a contradiction. Thus r₀ = 1. Then we have Ψ(g₂) - Ψ(g₁) = Ψ(g₁) - Ψ(g₀). Similarly, one can obtain that Ψ(g_i+1) - Ψ(g_i) = Ψ(g_i) - Ψ(g_i-1) for all i = 0, 1, ..., k - 1. Thus Ψ(g_i+1) - Ψ(g_i) = Ψ(g₁) - Ψ(g₀) for all i = 0, 1, ..., k - 1. Hence

Ψ ( f 1 ) - Ψ ( f 0 ) = Ψ ( g k ) - Ψ ( g 0 ) = Ψ ( g k ) - Ψ ( g k - 1 ) + Ψ ( g k - 1 ) - Ψ ( g k - 2 ) + ⋯ + Ψ ( g 1 ) - Ψ ( g 0 ) = k ( Ψ ( g 1 ) - Ψ ( g 0 ) ) .

Hence

| | Ψ ( f 1 ) - Ψ ( f 0 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , … , Ψ ( f n ) - Ψ ( f 0 ) | | β = | | k ( Ψ ( g 1 ) - Ψ ( g 0 ) ) , Ψ ( f 2 ) - Ψ ( f 0 ) , … , Ψ ( f n ) - Ψ ( f 0 ) | | β = | | k Ψ ( g 1 ) - Ψ ( g 0 ) , Ψ ( f 2 ) - Ψ ( f 0 ) , … , Ψ ( f n ) - Ψ ( f 0 ) | | β = k .