From now on, we use the symbol \(\| x \|\) to denote the norm of x defined by \(\| x \| := \sqrt{x_{1}^{2} + x_{2}^{2}}\) or \(\| x \| := \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}}\) for each \(x = (x_{1}, x_{2}) \in \mathbb{R}^{2}\) or \(x = (x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{3}\), respectively.
On bounded subsets of \(\mathbb{R}^{2}\)
To prove the next lemma we will use Lemma 2.2 to almost halve the number of unknowns to consider.
In the following lemma, the parameters \(c_{ij}\) are assumed to be positive real numbers, while they were assumed to be positive integers in [8].
Lemma 3.1
Let \(\{ e_{1}, e_{2} \}\) be the standard basis for the 2-dimensional Euclidean space \(\mathbb{R}^{2}\), let D be a subset of \(\mathbb{R}^{2}\) that includes 0, \(e_{1}\), and \(e_{2}\), and let \(f : D \to \mathbb{R}^{2}\) be a function that satisfies \(f(0) = 0\) and the inequality (1.3) for all \(x, y \in \{ 0, e_{1}, e_{2} \}\) and for some constant ε with \(0 < \varepsilon \leq \frac{1}{13}\). In view of Lemma 2.2, it can be assumed that \(f(e_{1}) = (e_{11}^{\prime }, 0)\) and \(f(e_{2}) = (e_{21}^{\prime }, e_{22}^{\prime })\), where \(e_{11}^{\prime }\geq 0\) and \(e_{22}^{\prime }\geq 0\). Then, there exist real numbers \(c_{ij} > 0\), \(i, j \in \{ 1, 2 \}\) with \(j \leq i\), such that
$$ \textstyle\begin{cases} -c_{ij} \varepsilon \leq e_{ij}^{\prime }\leq c_{ij} \varepsilon & ( \textit{for }i > j), \\ 1 - c_{ii} \varepsilon \leq e_{ii}^{\prime }\leq 1 + \varepsilon & ( \textit{for }i = j). \end{cases} $$
(3.1)
In particular, \(c_{11} = 1.0000000\), \(c_{21} \approx 3.7403981\), and \(c_{22} \approx 1.5978326\) may be selected.
Proof
In view of (1.3) and \(f(0) = 0\), we have
$$ \bigl\vert \bigl\Vert f(e_{1}) \bigr\Vert - 1 \bigr\vert \leq \varepsilon ,\qquad \bigl\vert \bigl\Vert f(e_{2}) \bigr\Vert - 1 \bigr\vert \leq \varepsilon , \qquad \bigl\vert \bigl\Vert f(e_{1}) - f(e_{2}) \bigr\Vert - \sqrt{2} \bigr\vert \leq \varepsilon $$
for any ε with \(0 < \varepsilon \leq \frac{1}{13}\). Hence, it follows from these inequalities that
$$\begin{aligned}& 1 - \varepsilon \leq e_{11}^{\prime }\leq 1 + \varepsilon , \end{aligned}$$
(3.2)
$$\begin{aligned}& (1 - \varepsilon )^{2} \leq e_{21}^{\prime 2} + e_{22}^{\prime 2} \leq (1 + \varepsilon )^{2}, \end{aligned}$$
(3.3)
$$\begin{aligned}& ( \sqrt{2} - \varepsilon )^{2} \leq \bigl(e_{11}^{\prime }- e_{21}^{\prime } \bigr)^{2} + e_{22}^{\prime 2} \leq ( \sqrt{2} + \varepsilon )^{2}. \end{aligned}$$
(3.4)
It follows from (3.4) that
$$ ( \sqrt{2} - \varepsilon )^{2} - e_{11}^{\prime 2} - \bigl( e_{21}^{ \prime 2} + e_{22}^{\prime 2} \bigr) \leq -2 e_{11}^{\prime }e_{21}^{\prime } \leq ( \sqrt{2} + \varepsilon )^{2} - e_{11}^{\prime 2} - \bigl( e_{21}^{\prime 2} + e_{22}^{\prime 2} \bigr). $$
We recall that \(0 < \varepsilon \leq \frac{1}{13}\). By (3.2) and (3.3), we obtain
$$ -\frac{26 + 13 \sqrt{2}}{12} \varepsilon \leq \frac{-(4 + 2 \sqrt{2}) \varepsilon + \varepsilon ^{2}}{2 (1 - \varepsilon )} \leq e_{21}^{\prime }\leq \frac{(4 + 2 \sqrt{2}) \varepsilon + \varepsilon ^{2}}{2 (1 - \varepsilon )} \leq \frac{53 + 26 \sqrt{2}}{24} \varepsilon . $$
Now, we select the smallest possible positive real number \(c_{21}\) that satisfies
$$ -c_{21} \varepsilon \leq -\frac{26 + 13 \sqrt{2}}{12} \varepsilon \leq e_{21}^{\prime }\leq \frac{53 + 26 \sqrt{2}}{24} \varepsilon \leq c_{21} \varepsilon . $$
Hence, we may choose
$$ c_{21} = \frac{53 + 26 \sqrt{2}}{24} \approx 3.7403981 $$
(3.5)
as the smallest possible positive number that satisfies the last inequality.
Furthermore, using our assumption that \(0 < \varepsilon \leq \frac{1}{13}\), it follows from (3.1) with \(i = 2\) and \(j = 1\), (3.3), and (3.5) that
$$ 1 - 2 \varepsilon - \frac{3585 + 2756 \sqrt{2}}{576} \varepsilon ^{2} = 1 - 2 \varepsilon + \bigl( 1 - c_{21}^{2} \bigr) \varepsilon ^{2} \leq e_{22}^{ \prime 2} \leq (1 + \varepsilon )^{2}. $$
By solving the following inequality, where \(c_{22}\) is unknown,
$$ (1 - c_{22} \varepsilon )^{2} \leq 1 - 2 \varepsilon - \frac{3585 + 2756 \sqrt{2}}{576} \varepsilon ^{2}, $$
we obtain
$$ \begin{aligned} & \frac{1}{\varepsilon } \biggl( 1 - \sqrt{1 - 2 \varepsilon - \frac{3585 + 2756 \sqrt{2}}{576} \varepsilon ^{2}} \biggr) \\ &\quad \leq c_{22} \leq \frac{1}{\varepsilon } \biggl( 1 + \sqrt{1 - 2 \varepsilon - \frac{3585 + 2756 \sqrt{2}}{576} \varepsilon ^{2}} \biggr). \end{aligned} $$
(3.6)
Thus, putting \(\varepsilon = \frac{1}{13}\) in the lower bound for \(c_{22}\) in (3.6), we may select
$$\begin{aligned} c_{22} = 13 - \sqrt{\frac{78\text{,}783 - 2756 \sqrt{2}}{576}} \approx 1.5978326 \end{aligned}$$
(3.7)
as the smallest possible positive constant that satisfies the inequality (3.6). □
In the following theorem, we set \(e_{1} = (1, 0)\) and \(e_{2} = (0, 1)\) such that \(\{ e_{1}, e_{2} \}\) is the standard basis for the 2-dimensional Euclidean space \(\mathbb{R}^{2}\). We denote by \(B_{d}(0)\) the closed ball of radius d and centered at the origin of \(\mathbb{R}^{2}\), i.e., \(B_{d}(0) = \{ x \in \mathbb{R}^{2} : \| x \| \leq d \}\).
Theorem 3.2
Let D be a subset of the Euclidean space \(\mathbb{R}^{2}\) such that \(\{ 0, e_{1}, e_{2} \} \subset D \subset B_{d}(0)\) for some \(d \geq 1\) and let \(f : D \to \mathbb{R}^{2}\) be a function that satisfies \(f(0) = 0\) and the inequality (1.3) for all \(x, y \in D\) and for some constant ε with \(0 < \varepsilon \leq \frac{1}{13}\). Then, there exists an isometry \(U : D \to \mathbb{R}^{2}\) such that
$$ \bigl\Vert f(x) - U(x) \bigr\Vert \leq (8d + 4) \varepsilon $$
(3.8)
for all \(x \in D\).
Proof
Due to Lemma 2.2, we may assume that \(f(e_{1}) = (e_{11}^{\prime }, 0)\) and \(f(e_{2}) = (e_{21}^{\prime }, e_{22}^{\prime })\) with \(e_{11}^{\prime }\geq 0\) and \(e_{22}^{\prime }\geq 0\). For every point \(x = (x_{1}, x_{2}) \in D\), let \(f(x) = x^{\prime }= (x_{1}^{\prime }, x_{2}^{\prime })\). Then, by (1.3), we have
$$\begin{aligned}& \bigl\vert \sqrt{x_{1}^{\prime 2} + x_{2}^{\prime 2}} - \sqrt{x_{1}^{2} + x_{2}^{2}} \bigr\vert \leq \varepsilon , \end{aligned}$$
(3.9)
$$\begin{aligned}& \bigl\vert \sqrt{ \bigl(x_{1}^{\prime }- e_{11}^{\prime } \bigr)^{2} + x_{2}^{\prime 2}} - \sqrt{(x_{1} - 1)^{2} + x_{2}^{2}} \bigr\vert \leq \varepsilon , \end{aligned}$$
(3.10)
$$\begin{aligned}& \bigl\vert \sqrt{ \bigl(x_{1}^{\prime }- e_{21}^{\prime } \bigr)^{2} + \bigl(x_{2}^{\prime }- e_{22}^{\prime } \bigr)^{2}} - \sqrt{x_{1}^{2} + (x_{2} - 1)^{2}} \bigr\vert \leq \varepsilon . \end{aligned}$$
(3.11)
According to (3.9), we obtain
$$\begin{aligned} \begin{aligned} & \bigl\vert \bigl( x_{1}^{\prime 2} + x_{2}^{\prime 2} \bigr) - \bigl( x_{1}^{2} + x_{2}^{2} \bigr) \bigr\vert \\ &\quad = \bigl\vert \sqrt{x_{1}^{\prime 2} + x_{2}^{\prime 2}} - \sqrt{x_{1}^{2} + x_{2}^{2}} \bigr\vert \bigl\vert \sqrt{x_{1}^{\prime 2} + x_{2}^{\prime 2}} + \sqrt{x_{1}^{2} + x_{2}^{2}} \bigr\vert \\ &\quad \leq \varepsilon (d + \varepsilon + d) \leq \biggl( 2d + \frac{1}{13} \biggr) \varepsilon , \end{aligned} \end{aligned}$$
(3.12)
since \(\sqrt{x_{1}^{\prime 2} + x_{2}^{\prime 2}} = \| f(x) \| \leq \| x \| + \varepsilon \leq d + \varepsilon \), \(\sqrt{x_{1}^{2} + x_{2}^{2}} \leq d\) and \(0 < \varepsilon \leq \frac{1}{13}\). Similarly, by (3.10), we obtain
$$ \bigl\vert \bigl( \bigl( x_{1}^{\prime }- e_{11}^{\prime } \bigr)^{2} + x_{2}^{ \prime 2} \bigr) - \bigl( (x_{1} - 1)^{2} + x_{2}^{2} \bigr) \bigr\vert \leq \biggl( 2d + \frac{27}{13} \biggr) \varepsilon , $$
(3.13)
since \(\sqrt{(x_{1}^{\prime }- e_{11}^{\prime })^{2} + x_{2}^{\prime 2}} = \| f(x) - f(e_{1}) \| \leq \| x - e_{1} \| + \varepsilon \leq d + 1 + \varepsilon \), \(\sqrt{(x_{1} - 1)^{2} + x_{2}^{2}} = \| x - e_{1} \| \leq d + 1\) and \(0 < \varepsilon \leq \frac{1}{13}\). Analogously, in view of (3.11), we have
$$ \bigl\vert \bigl( \bigl( x_{1}^{\prime }- e_{21}^{\prime } \bigr)^{2} + \bigl( x_{2}^{\prime }- e_{22}^{\prime } \bigr)^{2} \bigr) - \bigl( x_{1}^{2} + (x_{2} - 1)^{2} \bigr) \bigr\vert \leq \biggl( 2d + \frac{27}{13} \biggr) \varepsilon , $$
(3.14)
since \(\sqrt{(x_{1}^{\prime }- e_{21}^{\prime })^{2} + (x_{2}^{\prime }- e_{22}^{\prime })^{2}} = \| f(x) - f(e_{2}) \| \leq \| x - e_{2} \| + \varepsilon \leq d + 1 + \varepsilon \) and \(0 < \varepsilon \leq \frac{1}{13}\).
It now follows from (3.13) that
$$ \bigl\vert \bigl( x_{1}^{\prime 2} + x_{2}^{\prime 2} \bigr) - \bigl( x_{1}^{2} + x_{2}^{2} \bigr) - 2 e_{11}^{\prime }x_{1}^{\prime }+ 2 x_{1} + e_{11}^{ \prime 2} - 1 \bigr\vert \leq \biggl( 2d + \frac{27}{13} \biggr) \varepsilon . $$
Using (3.12), we obtain
$$ - \biggl( 4d + \frac{28}{13} \biggr) \varepsilon \leq -2 e_{11}^{\prime }x_{1}^{\prime }+ 2 x_{1} + e_{11}^{\prime 2} - 1 \leq \biggl( 4d + \frac{28}{13} \biggr) \varepsilon . $$
(3.15)
Similarly, using (3.14), we obtain
$$ \bigl\vert \bigl( x_{1}^{\prime 2} + x_{2}^{\prime 2} \bigr) - \bigl( x_{1}^{2} + x_{2}^{2} \bigr) - 2 e_{21}^{\prime }x_{1}^{\prime }- 2 e_{22}^{\prime }x_{2}^{\prime }+ 2 x_{2} + e_{21}^{\prime 2} + e_{22}^{\prime 2} - 1 \bigr\vert \leq \biggl( 2d + \frac{27}{13} \biggr) \varepsilon . $$
Using (3.12) again, we obtain
$$ - \biggl( 4d + \frac{28}{13} \biggr) \varepsilon \leq -2 e_{21}^{\prime }x_{1}^{\prime }- 2 e_{22}^{\prime }x_{2}^{\prime }+ 2 x_{2} + e_{21}^{ \prime 2} + e_{22}^{\prime 2} - 1 \leq \biggl( 4d + \frac{28}{13} \biggr) \varepsilon . $$
(3.16)
Moreover, put \(x = e_{1}\) and \(y = 0\) in (1.3) and use (3.1) to obtain
$$ \bigl\vert e_{11}^{\prime }- 1 \bigr\vert \leq \varepsilon \quad \mbox{and}\quad \bigl\vert e_{11}^{\prime 2} - 1 \bigr\vert = \bigl\vert e_{11}^{\prime }- 1 \bigr\vert \bigl\vert e_{11}^{\prime }+ 1 \bigr\vert \leq \frac{27}{13} \varepsilon . $$
Hence, it follows from (3.15) that
$$ \bigl\vert x_{1} - x_{1}^{\prime } \bigr\vert \leq \biggl( 3d + \frac{57}{26} \biggr) \varepsilon , $$
(3.17)
since \(\max \{ | x_{1}^{\prime }|, | x_{2}^{\prime }| \} \leq \| f(x) \| \leq \| x \| + \varepsilon \leq d + \frac{1}{13}\) and
$$ -2 e_{11}^{\prime }x_{1}^{\prime }+ 2 x_{1} + e_{11}^{\prime 2} - 1 = 2 \bigl( 1 - e_{11}^{\prime } \bigr) x_{1}^{\prime }+ 2 \bigl( x_{1} - x_{1}^{\prime } \bigr) + e_{11}^{\prime 2} - 1. $$
On the other hand, it follows from (3.1) that
$$ \bigl\vert 2 e_{21}^{\prime }x_{1}^{\prime } \bigr\vert \leq 2 c_{21} \bigl\vert x_{1}^{\prime } \bigr\vert \varepsilon \leq 2 c_{21} \biggl( d + \frac{1}{13} \biggr) \varepsilon . $$
Due to (3.1), we obtain
$$ -\varepsilon \leq 1 - e_{22}^{\prime }\leq c_{22} \varepsilon . $$
By (3.16) together with (3.3), we obtain
$$ - \biggl( (2 + c_{21}) d + \frac{55 + 2 c_{21}}{26} \biggr) \varepsilon \leq x_{2} - e_{22}^{\prime }x_{2}^{\prime } \leq \biggl( (2 + c_{21}) d + \frac{27 + c_{21}}{13} \biggr) \varepsilon , $$
or since \(x_{2} - e_{22}^{\prime }x_{2}^{\prime }= (x_{2} - x_{2}^{\prime }) + (1 - e_{22}^{\prime }) x_{2}^{\prime }\),
$$\begin{aligned}& - \biggl( (2 + c_{21} + c_{22}) d + \frac{55 + 2 c_{21} + 2 c_{22}}{26} \biggr) \varepsilon \\& \quad \leq x_{2} - x_{2}^{\prime }\leq \biggl( (2 + c_{21} + c_{22}) d + \frac{27 + c_{21} + c_{22}}{13} \biggr) \varepsilon . \end{aligned}$$
Therefore, we have
$$ \bigl\vert x_{2} - x_{2}^{\prime } \bigr\vert \leq \biggl( (2 + c_{21} + c_{22}) d + \frac{55 + 2 c_{21} + 2 c_{22}}{26} \biggr) \varepsilon . $$
(3.18)
Finally, we define an isometry \(U : D \to \mathbb{R}^{2}\) by \(U(x) = x = (x_{1}, x_{2})\). It then follows from (3.17), (3.18) and Lemma 3.1 that
$$ \begin{aligned} \bigl\Vert f(x) - U(x) \bigr\Vert & = \bigl\Vert f(x) - x \bigr\Vert = \bigl\Vert \bigl(x_{1}^{\prime }- x_{1}, x_{2}^{\prime }- x_{2} \bigr) \bigr\Vert \\ & = \sqrt{ \bigl( x_{1}^{\prime }- x_{1} \bigr)^{2} + \bigl( x_{2}^{\prime }- x_{2} \bigr)^{2}} \\ & \leq (8d + 4) \varepsilon , \end{aligned} $$
which completes our proof. □
By \(c_{ij}\) in Lemma 3.1, (3.17) and (3.18), we can express the upper bound of inequality (3.8) more precisely using rational numbers as:
$$\begin{aligned} \begin{aligned} \bigl\Vert f(x) - U(x) \bigr\Vert & = \sqrt{62.8496299 d^{2} + 50.2268490 d + 11.1869791} \varepsilon \\ & \leq \sqrt{62.8496299 d^{2} + 51.0138281 d + 10.4000000} \varepsilon \\ & \leq (7.9277759 d + 3.2249031) \varepsilon . \end{aligned} \end{aligned}$$
On bounded subsets of \(\mathbb{R}^{3}\)
Lemma 3.1 can now be extended without difficulty to the case of the 3-dimensional Euclidean space. It is evident that there exist positive real numbers \(c_{ij}\), \(i, j \in \{ 1, 2, 3 \}\) with \(j \leq i\) that satisfy the conditions in (3.1) whenever the function f satisfies \(f(0) = 0\) and the inequality (1.3) for all \(x, y \in \{ 0, e_{1}, e_{2}, e_{3} \}\).
In the following lemma, the parameters \(c_{ij}\) are assumed to be positive real numbers, while they were assumed to be positive integers in [8].
Lemma 3.3
Let \(\{ e_{1}, e_{2}, e_{3} \}\) be the standard basis for the 3-dimensional Euclidean space \(\mathbb{R}^{3}\), let D be a subset of \(\mathbb{R}^{3}\) that satisfies \(\{ 0, e_{1}, e_{2}, e_{3} \} \subset D\), and let \(f : D \to \mathbb{R}^{3}\) be a function that satisfies \(f(0) = 0\) and the inequality (1.3) for all \(x, y \in \{ 0, e_{1}, e_{2}, e_{3} \}\) and for some constant ε with \(0 < \varepsilon \leq \frac{1}{13}\). By Lemma 2.2, it can be assumed that \(f(e_{1}) = (e_{11}^{\prime }, 0, 0)\), \(f(e_{2}) = (e_{21}^{\prime }, e_{22}^{\prime }, 0)\), and \(f(e_{3}) = (e_{31}^{\prime }, e_{32}^{\prime }, e_{33}^{\prime })\), where \(e_{11}^{\prime }\geq 0\), \(e_{22}^{\prime }\geq 0\), and \(e_{33}^{\prime }\geq 0\). Then, there exist positive real numbers \(c_{ij}\), \(i, j \in \{ 1, 2, 3 \}\) with \(j \leq i\) that satisfy the inequalities in (3.1). In particular, \(c_{11} = 1.0000000\), \(c_{21} \approx 3.7403981\), \(c_{22} \approx 1.5978326\), \(c_{31} \approx 3.7403981\), \(c_{32} \approx 5.1635231\), and \(c_{33} \approx 2.8340052\) may be selected.
Proof
In view of (1.3) and \(f(0) = 0\), we have
$$ \begin{aligned} &\bigl\vert \bigl\Vert f(e_{1}) \bigr\Vert - 1 \bigr\vert \leq \varepsilon , \qquad \bigl\vert \bigl\Vert f(e_{2}) \bigr\Vert - 1 \bigr\vert \leq \varepsilon ,\qquad \bigl\vert \bigl\Vert f(e_{3}) \bigr\Vert - 1 \bigr\vert \leq \varepsilon , \\ &\bigl\vert \bigl\Vert f(e_{1}) - f(e_{2}) \bigr\Vert - \sqrt{2} \bigr\vert \leq \varepsilon ,\qquad \bigl\vert \bigl\Vert f(e_{2}) - f(e_{3}) \bigr\Vert - \sqrt{2} \bigr\vert \leq \varepsilon , \\ &\bigl\vert \bigl\Vert f(e_{3}) - f(e_{1}) \bigr\Vert - \sqrt{2} \bigr\vert \leq \varepsilon \end{aligned} $$
for any ε with \(0 < \varepsilon \leq \frac{1}{13}\). Therefore, from the inequalities above, we obtain the following inequalities along with (3.2), (3.3) and (3.4):
$$\begin{aligned}& (1 - \varepsilon )^{2} \leq e_{31}^{\prime 2} + e_{32}^{\prime 2} + e_{33}^{ \prime 2} \leq (1 + \varepsilon )^{2}, \end{aligned}$$
(3.19)
$$\begin{aligned}& ( \sqrt{2} - \varepsilon )^{2} \leq \bigl(e_{21}^{\prime }- e_{31}^{\prime } \bigr)^{2} + \bigl(e_{22}^{\prime }- e_{32}^{\prime } \bigr)^{2} + e_{33}^{\prime 2} \leq ( \sqrt{2} + \varepsilon )^{2}, \end{aligned}$$
(3.20)
$$\begin{aligned}& ( \sqrt{2} - \varepsilon )^{2} \leq \bigl(e_{31}^{\prime }- e_{11}^{\prime } \bigr)^{2} + e_{32}^{\prime 2} + e_{33}^{\prime 2} \leq ( \sqrt{2} + \varepsilon )^{2}. \end{aligned}$$
(3.21)
Moreover, using (3.21), we have
$$ ( \sqrt{2} - \varepsilon )^{2} - e_{11}^{\prime 2} - \bigl( e_{31}^{ \prime 2} + e_{32}^{\prime 2} + e_{33}^{\prime 2} \bigr) \leq -2 e_{11}^{\prime }e_{31}^{\prime } \leq ( \sqrt{2} + \varepsilon )^{2} - e_{11}^{ \prime 2} - \bigl( e_{31}^{\prime 2} + e_{32}^{\prime 2} + e_{33}^{ \prime 2} \bigr). $$
In view of (3.2) and (3.19), we obtain
$$ \frac{-(4 + 2 \sqrt{2}) \varepsilon + \varepsilon ^{2}}{2 (1 - \varepsilon )} \leq e_{31}^{\prime }\leq \frac{(4 + 2 \sqrt{2}) \varepsilon + \varepsilon ^{2}}{2 (1 - \varepsilon )} $$
and we solve the following inequalities
$$ -c_{31} \varepsilon \leq \frac{-(4 + 2 \sqrt{2}) \varepsilon + \varepsilon ^{2}}{2 (1 - \varepsilon )}\quad \mbox{and}\quad \frac{(4 + 2 \sqrt{2}) \varepsilon + \varepsilon ^{2}}{2 (1 - \varepsilon )} \leq c_{31} \varepsilon $$
and we find
$$ c_{31} = \frac{53 + 26 \sqrt{2}}{24} \approx 3.7403981 $$
(3.22)
as the smallest possible positive real number that satisfies the preceding inequalities.
By (3.20), we obtain
$$ \begin{aligned} &2 - 2 \sqrt{2} \varepsilon + \varepsilon ^{2} - \bigl( e_{21}^{\prime 2} + e_{22}^{\prime 2} \bigr) - \bigl( e_{31}^{\prime 2} + e_{32}^{\prime 2} + e_{33}^{\prime 2} \bigr) \\ &\quad \leq -2 e_{21}^{\prime }e_{31}^{\prime }- 2 e_{22}^{\prime }e_{32}^{\prime }\leq 2 + 2 \sqrt{2} \varepsilon + \varepsilon ^{2} - \bigl( e_{21}^{\prime 2} + e_{22}^{\prime 2} \bigr) - \bigl( e_{31}^{\prime 2} + e_{32}^{\prime 2} + e_{33}^{\prime 2} \bigr). \end{aligned} $$
Moreover, we use (3.3) and (3.19) to obtain
$$ - ( 4 + 2 \sqrt{2} ) \varepsilon + \varepsilon ^{2} - 2 e_{21}^{\prime }e_{31}^{\prime }\leq 2 e_{22}^{\prime }e_{32}^{\prime }\leq ( 4 + 2 \sqrt{2} ) \varepsilon + \varepsilon ^{2} - 2 e_{21}^{\prime }e_{31}^{\prime }, $$
or one time using (3.1) with \(i = 2\) and \(j = 1\) and the next time using (3.1) with \(i = 3\) and \(j = 1\),
$$ - ( 4 + 2 \sqrt{2} ) \varepsilon + \varepsilon ^{2} - 2 c_{21} c_{31} \varepsilon ^{2} \leq 2 e_{22}^{\prime }e_{32}^{\prime }\leq ( 4 + 2 \sqrt{2} ) \varepsilon + \varepsilon ^{2} + 2 c_{21} c_{31} \varepsilon ^{2}. $$
Due to (3.1) with \(i = j = 2\), we have
$$ \frac{- ( 4 + 2 \sqrt{2} ) \varepsilon + \varepsilon ^{2} - 2 c_{21} c_{31} \varepsilon ^{2}}{2 (1 - c_{22} \varepsilon )} \leq e_{32}^{\prime }\leq \frac{ ( 4 + 2 \sqrt{2} ) \varepsilon + \varepsilon ^{2} + 2 c_{21} c_{31} \varepsilon ^{2}}{2 (1 - c_{22} \varepsilon )} $$
and we solve the following inequalities
$$ -c_{32} \varepsilon \leq \frac{- ( 4 + 2 \sqrt{2} ) \varepsilon + \varepsilon ^{2} - 2 c_{21} c_{31} \varepsilon ^{2}}{2 (1 - c_{22} \varepsilon )}\quad \mbox{and}\quad \frac{ ( 4 + 2 \sqrt{2} ) \varepsilon + \varepsilon ^{2} + 2 c_{21} c_{31} \varepsilon ^{2}}{2 (1 - c_{22} \varepsilon )} \leq c_{32} \varepsilon . $$
We use (3.5), (3.7), and (3.22) and put \(\varepsilon = \frac{1}{13}\) in the second inequality to obtain
$$ c_{32} = \frac{4 + 2 \sqrt{2} + \frac{1}{13} (1 + 2 \times 3.7403981^{2})}{2 (1 - \frac{1}{13} \times 1.5978326)} \approx 5.1635231 $$
(3.23)
as the smallest possible positive number that satisfies the last inequalities.
Finally, by (3.1) with \(i = 3\) and \(j = 1\), (3.1) with \(i = 3\) and \(j = 2\), and by (3.19), we have
$$ 1 - 2 \varepsilon - \bigl( c_{31}^{2} + c_{32}^{2} - 1 \bigr) \varepsilon ^{2} \leq e_{33}^{\prime 2} \leq (1 + \varepsilon )^{2}. $$
Moreover, we solve the following inequality
$$ (1 - c_{33} \varepsilon )^{2} \leq 1 - 2 \varepsilon - \bigl( c_{31}^{2} + c_{32}^{2} - 1 \bigr) \varepsilon ^{2}, $$
whose solution is given as
$$ \frac{1}{\varepsilon } \bigl( 1 - \sqrt{1 - 2 \varepsilon - \bigl( c_{31}^{2} + c_{32}^{2} - 1 \bigr) \varepsilon ^{2}} \bigr) \leq c_{33} \leq \frac{1}{\varepsilon } \bigl( 1 + \sqrt{1 - 2 \varepsilon - \bigl( c_{31}^{2} + c_{32}^{2} - 1 \bigr) \varepsilon ^{2}} \bigr). $$
In addition, we use (3.22) and (3.23) and put \(\varepsilon = \frac{1}{13}\) in the lower bound for \(c_{33}\) to obtain
$$ c_{33} = 13 - \sqrt{144 - c_{31}^{2} - c_{32}^{2}} \approx 2.8340052 $$
(3.24)
as small a positive real number as possible that satisfies the last inequalities. □
In the following theorem, we set \(e_{1} = (1, 0, 0)\), \(e_{2} = (0, 1, 0)\), and \(e_{3} = (0, 0, 1)\) such that \(\{ e_{1}, e_{2}, e_{3} \}\) is the standard basis for the 3-dimensional Euclidean space \(\mathbb{R}^{3}\). We denote by \(B_{d}(0)\) the closed ball of radius d and centered at the origin of \(\mathbb{R}^{3}\), i.e., \(B_{d}(0) = \{ x \in \mathbb{R}^{3} : \| x \| \leq d \}\).
Theorem 3.4
Let D be a subset of the 3-dimensional Euclidean space \(\mathbb{R}^{3}\) such that \(\{ 0, e_{1}, e_{2}, e_{3} \} \subset D \subset B_{d}(0)\) for some \(d \geq 1\) and let \(f : D \to \mathbb{R}^{3}\) be a function that satisfies \(f(0) = 0\) and the inequality (1.3) for all \(x, y \in D\) and for some constant ε with \(0 < \varepsilon \leq \frac{1}{13}\). Then, there exists an isometry \(U : D \to \mathbb{R}^{3}\) such that
$$ \bigl\Vert f(x) - U(x) \bigr\Vert \leq (16d + 5) \varepsilon $$
(3.25)
for all \(x \in D\).
Proof
Considering Lemma 2.2, we can assume that \(f(e_{1}) = (e_{11}^{\prime }, 0, 0)\), \(f(e_{2}) = (e_{21}^{\prime }, e_{22}^{\prime }, 0)\), and \(f(e_{3}) = (e_{31}^{\prime }, e_{32}^{\prime }, e_{33}^{\prime })\), where \(e_{11}^{\prime }\geq 0\), \(e_{22}^{\prime }\geq 0\), and \(e_{33}^{\prime }\geq 0\).
For any point \(x = (x_{1}, x_{2}, x_{3})\) of D, let \(f(x) = x^{\prime }= (x_{1}^{\prime }, x_{2}^{\prime }, x_{3}^{\prime })\). It then follows from (1.3) that
$$\begin{aligned}& \bigl\vert \sqrt{x_{1}^{\prime 2} + x_{2}^{\prime 2} + x_{3}^{\prime 2}} - \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}} \bigr\vert \leq \varepsilon , \end{aligned}$$
(3.26)
$$\begin{aligned}& \bigl\vert \sqrt{ \bigl(x_{1}^{\prime }- e_{11}^{\prime } \bigr)^{2} + x_{2}^{\prime 2} + x_{3}^{\prime 2}} - \sqrt{(x_{1} - 1)^{2} + x_{2}^{2} + x_{3}^{2}} \bigr\vert \leq \varepsilon , \end{aligned}$$
(3.27)
$$\begin{aligned}& \bigl\vert \sqrt{ \bigl(x_{1}^{\prime }- e_{21}^{\prime } \bigr)^{2} + \bigl(x_{2}^{\prime }- e_{22}^{\prime } \bigr)^{2} + x_{3}^{\prime 2}} - \sqrt{x_{1}^{2} + (x_{2} - 1)^{2} + x_{3}^{2}} \bigr\vert \leq \varepsilon , \end{aligned}$$
(3.28)
$$\begin{aligned}& \bigl\vert \sqrt{ \bigl(x_{1}^{\prime }- e_{31}^{\prime } \bigr)^{2} + \bigl(x_{2}^{\prime }- e_{32}^{\prime } \bigr)^{2} + \bigl(x_{3}^{\prime }- e_{33}^{\prime } \bigr)^{2}} - \sqrt{x_{1}^{2} + x_{2}^{2} + (x_{3} - 1)^{2}} \bigr\vert \leq \varepsilon . \end{aligned}$$
(3.29)
It follows from (3.26) that
$$\begin{aligned} \begin{aligned} & \bigl\vert \bigl( x_{1}^{\prime 2} + x_{2}^{\prime 2} + x_{3}^{ \prime 2} \bigr) - \bigl( x_{1}^{2} + x_{2}^{2} + x_{3}^{2} \bigr) \bigr\vert \\ &\quad = \bigl\vert \sqrt{x_{1}^{\prime 2} + x_{2}^{\prime 2} + x_{3}^{\prime 2}} - \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}} \bigr\vert \bigl\vert \sqrt{x_{1}^{ \prime 2} + x_{2}^{\prime 2} + x_{3}^{\prime 2}} + \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}} \bigr\vert \\ &\quad \leq \varepsilon (d + \varepsilon + d) \leq \biggl( 2d + \frac{1}{13} \biggr) \varepsilon , \end{aligned} \end{aligned}$$
(3.30)
since \(\sqrt{x_{1}^{\prime 2} + x_{2}^{\prime 2} + x_{3}^{\prime 2}} = \| f(x) \| \leq \| x \| + \varepsilon \leq d + \varepsilon \), \(\sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}} \leq d\) and \(0 < \varepsilon \leq \frac{1}{13}\). Similarly, it follows from (3.27) that
$$ \bigl\vert \bigl( \bigl( x_{1}^{\prime }- e_{11}^{\prime } \bigr)^{2} + x_{2}^{ \prime 2} + x_{3}^{\prime 2} \bigr) - \bigl( (x_{1} - 1)^{2} + x_{2}^{2} + x_{3}^{2} \bigr) \bigr\vert \leq \biggl( 2d + \frac{27}{13} \biggr) \varepsilon , $$
(3.31)
since \(\sqrt{(x_{1}^{\prime }- e_{11}^{\prime })^{2} + x_{2}^{\prime 2} + x_{3}^{ \prime 2}} = \| f(x) - f(e_{1}) \| \leq \| x - e_{1} \| + \varepsilon \leq d + 1 + \varepsilon \), \(\sqrt{(x_{1} - 1)^{2} + x_{2}^{2} + x_{3}^{2}} = \| x - e_{1} \| \leq d + 1\) and \(0 < \varepsilon \leq \frac{1}{13}\). Analogously, by (3.28) and (3.29), we obtain
$$ \bigl\vert \bigl( \bigl( x_{1}^{\prime }- e_{21}^{\prime } \bigr)^{2} + \bigl( x_{2}^{\prime }- e_{22}^{\prime } \bigr)^{2} + x_{3}^{\prime 2} \bigr) - \bigl( x_{1}^{2} + (x_{2} - 1)^{2} + x_{3}^{2} \bigr) \bigr\vert \leq \biggl( 2d + \frac{27}{13} \biggr) \varepsilon $$
(3.32)
and
$$ \bigl\vert \bigl( \bigl( x_{1}^{\prime }- e_{31}^{\prime } \bigr)^{2} + \bigl( x_{2}^{\prime }- e_{32}^{\prime } \bigr)^{2} + \bigl( x_{3}^{\prime }- e_{33}^{\prime } \bigr)^{2} \bigr) - \bigl( x_{1}^{2} + x_{2}^{2} + (x_{3} - 1)^{2} \bigr) \bigr\vert \leq \biggl( 2d + \frac{27}{13} \biggr) \varepsilon , $$
(3.33)
since \(\sqrt{x_{1}^{2} + (x_{2} - 1)^{2} + x_{3}^{2}} \leq d + 1\), \(\sqrt{x_{1}^{2} + x_{2}^{2} + (x_{3} - 1)^{2}} \leq d + 1\), and \(0 < \varepsilon \leq \frac{1}{13}\).
It follows from (3.31) that
$$ \bigl\vert \bigl( x_{1}^{\prime 2} + x_{2}^{\prime 2} + x_{3}^{\prime 2} \bigr) - \bigl( x_{1}^{2} + x_{2}^{2} + x_{3}^{2} \bigr) - 2 e_{11}^{\prime }x_{1}^{\prime }+ 2 x_{1} + e_{11}^{\prime 2} - 1 \bigr\vert \leq \biggl( 2d + \frac{27}{13} \biggr) \varepsilon . $$
By using (3.30), we obtain
$$ - \biggl( 4d + \frac{28}{13} \biggr) \varepsilon \leq -2 e_{11}^{\prime }x_{1}^{\prime }+ 2 x_{1} + e_{11}^{\prime 2} - 1 \leq \biggl( 4d + \frac{28}{13} \biggr) \varepsilon . $$
(3.34)
Similarly, it follows from (3.32) that
$$ \bigl\vert \bigl( x_{1}^{\prime 2} + x_{2}^{\prime 2} + x_{3}^{\prime 2} \bigr) - \bigl( x_{1}^{2} + x_{2}^{2} + x_{3}^{2} \bigr) - 2 e_{21}^{\prime }x_{1}^{\prime }- 2 e_{22}^{\prime }x_{2}^{\prime }+ 2 x_{2} + e_{21}^{ \prime 2} + e_{22}^{\prime 2} - 1 \bigr\vert \leq \biggl( 2d + \frac{27}{13} \biggr) \varepsilon . $$
Using (3.30), we obtain
$$ - \biggl( 4d + \frac{28}{13} \biggr) \varepsilon \leq -2 e_{21}^{\prime }x_{1}^{\prime }- 2 e_{22}^{\prime }x_{2}^{\prime }+ 2 x_{2} + e_{21}^{ \prime 2} + e_{22}^{\prime 2} - 1 \leq \biggl( 4d + \frac{28}{13} \biggr) \varepsilon . $$
(3.35)
Analogously, we use (3.30) and (3.33) to obtain
$$ \begin{aligned} - \biggl( 4d + \frac{28}{13} \biggr) \varepsilon &\leq -2 e_{31}^{\prime }x_{1}^{\prime }- 2 e_{32}^{\prime }x_{2}^{\prime }- 2 e_{33}^{\prime }x_{3}^{\prime }+ 2 x_{3} + e_{31}^{\prime 2} + e_{32}^{\prime 2} + e_{33}^{ \prime 2} - 1 \\ &\leq \biggl( 4d + \frac{28}{13} \biggr) \varepsilon . \end{aligned} $$
(3.36)
Moreover, putting \(x = e_{1}\) and \(y = 0\) in (1.3) and using (3.1), we have
$$ \bigl\vert e_{11}^{\prime }- 1 \bigr\vert \leq \varepsilon \quad \mbox{and}\quad \bigl\vert e_{11}^{\prime 2} - 1 \bigr\vert = \bigl\vert e_{11}^{\prime }- 1 \bigr\vert \bigl\vert e_{11}^{\prime }+ 1 \bigr\vert \leq \frac{27}{13} \varepsilon . $$
Therefore, it follows from (3.34) that
$$ \bigl\vert x_{1} - x_{1}^{\prime } \bigr\vert \leq \biggl( 3d + \frac{57}{26} \biggr) \varepsilon , $$
(3.37)
since \(\max \{ | x_{1}^{\prime }|, | x_{2}^{\prime }|, | x_{3}^{\prime }| \} \leq \| f(x) \| \leq \| x \| + \varepsilon \leq d + \frac{1}{13}\) and
$$ -2 e_{11}^{\prime }x_{1}^{\prime }+ 2 x_{1} + e_{11}^{\prime 2} - 1 = 2 \bigl( 1 - e_{11}^{\prime } \bigr) x_{1}^{\prime }+ 2 \bigl( x_{1} - x_{1}^{\prime } \bigr) + e_{11}^{\prime 2} - 1. $$
On the other hand, it follows from (3.1) that
$$ \bigl\vert 2 e_{21}^{\prime }x_{1}^{\prime } \bigr\vert \leq 2 c_{21} \bigl\vert x_{1}^{\prime } \bigr\vert \varepsilon \leq 2 c_{21} \biggl( d + \frac{1}{13} \biggr) \varepsilon . $$
By (3.1), we obtain
$$ -\varepsilon \leq 1 - e_{22}^{\prime }\leq c_{22} \varepsilon . $$
Furthermore, using (3.35) together with (3.3), we have
$$ - \biggl( (2 + c_{21}) d + \frac{55 + 2 c_{21}}{26} \biggr) \varepsilon \leq x_{2} - e_{22}^{\prime }x_{2}^{\prime } \leq \biggl( (2 + c_{21}) d + \frac{27 + c_{21}}{13} \biggr) \varepsilon $$
or
$$\begin{aligned}& - \biggl( (2 + c_{21} + c_{22}) d + \frac{55 + 2 c_{21} + 2 c_{22}}{26} \biggr) \varepsilon \\& \quad \leq x_{2} - x_{2}^{\prime }\leq \biggl( (2 + c_{21} + c_{22}) d + \frac{27 + c_{21} + c_{22}}{13} \biggr) \varepsilon , \end{aligned}$$
since
$$ x_{2} - e_{22}^{\prime }x_{2}^{\prime }= \bigl( x_{2} - x_{2}^{\prime } \bigr) + \bigl( 1 - e_{22}^{\prime } \bigr) x_{2}^{\prime }. $$
Thus, we see that
$$ \bigl\vert x_{2} - x_{2}^{\prime } \bigr\vert \leq \biggl( (2 + c_{21} + c_{22}) d + \frac{55 + 2 c_{21} + 2 c_{22}}{26} \biggr) \varepsilon . $$
(3.38)
Analogously, by using (3.1), we obtain
$$ \bigl\vert 2 e_{31}^{\prime }x_{1}^{\prime } \bigr\vert \leq 2 c_{31} \biggl( d + \frac{1}{13} \biggr) \varepsilon\quad \mbox{and} \quad \bigl\vert 2 e_{32}^{\prime }x_{2}^{\prime } \bigr\vert \leq 2 c_{32} \biggl( d + \frac{1}{13} \biggr) \varepsilon . $$
Moreover, due to (3.36), we obtain
$$\begin{aligned}& - \biggl( 4d + \frac{28}{13} \biggr) \varepsilon + 2 e_{31}^{\prime }x_{1}^{\prime }+ 2 e_{32}^{\prime }x_{2}^{\prime }- \bigl( e_{31}^{ \prime 2} + e_{32}^{\prime 2} + e_{33}^{\prime 2} \bigr) + 1 \\& \quad \leq 2 \bigl( x_{3} - e_{33}^{\prime }x_{3}^{\prime } \bigr) \leq \biggl( 4d + \frac{28}{13} \biggr) \varepsilon + 2 e_{31}^{\prime }x_{1}^{\prime }+ 2 e_{32}^{\prime }x_{2}^{\prime }- \bigl( e_{31}^{\prime 2} + e_{32}^{ \prime 2} + e_{33}^{\prime 2} \bigr) + 1. \end{aligned}$$
In view of (3.19), we have
$$\begin{aligned}& - \biggl( (4 + 2 c_{31} + 2 c_{32}) d + \frac{55 + 2 c_{31} + 2 c_{32}}{13} \biggr) \varepsilon \\& \quad \leq 2 \bigl( x_{3} - e_{33}^{\prime }x_{3}^{\prime } \bigr) \leq \biggl( (4 + 2 c_{31} + 2 c_{32}) d + \frac{54 + 2 c_{31} + 2 c_{32}}{13} \biggr) \varepsilon . \end{aligned}$$
By (3.1), we see that \(1 - c_{33} \varepsilon \leq e_{33}^{\prime }\leq 1 + \varepsilon \) and hence,
$$ \bigl\vert x_{3} - x_{3}^{\prime } \bigr\vert \leq \biggl( (2 + c_{31} + c_{32} + c_{33}) d + \frac{55 + 2 c_{31} + 2 c_{32} + 2 c_{33}}{26} \biggr) \varepsilon . $$
(3.39)
Finally, we may define an isometry \(U : D \to \mathbb{R}^{3}\) by \(U(x) = x = (x_{1}, x_{2}, x_{3})\). Then, we use (3.37), (3.38), (3.39), and Lemma 3.3 to obtain
$$ \begin{aligned} \bigl\Vert f(x) - U(x) \bigr\Vert & = \bigl\Vert f(x) - x \bigr\Vert = \bigl\Vert \bigl(x_{1}^{\prime }- x_{1}, x_{2}^{\prime }- x_{2}, x_{3}^{\prime }- x_{3} \bigr) \bigr\Vert \\ & = \sqrt{ \bigl( x_{1}^{\prime }- x_{1} \bigr)^{2} + \bigl( x_{2}^{\prime }- x_{2} \bigr)^{2} + \bigl( x_{3}^{\prime }- x_{3} \bigr)^{2}} \\ & \leq \sqrt{252 d^{2} + 134 d + 21} \varepsilon \\ & \leq (16d + 5) \varepsilon , \end{aligned} $$
which completes our proof. □
By \(c_{ij}\) in Lemma 3.3, (3.37), (3.38), and (3.39), we can express the upper bound of inequality (3.25) more precisely using rational numbers as:
$$ \begin{aligned} \bigl\Vert f(x) - U(x) \bigr\Vert & = \sqrt{251.5802517 d^{2} + 133.1572732 d + 20.2971267} \varepsilon \\ & \leq \sqrt{251.5802517 d^{2} + 134.9643999 d + 18.4900000} \varepsilon \\ & \leq (15.8612816 d + 4.3000000) \varepsilon . \end{aligned} $$
On the other hand, it follows from [8] that
$$ \begin{aligned} \bigl\Vert f(x) - U(x) \bigr\Vert & = \sqrt{451 d^{2} + 1026 d + 587} \varepsilon \\ & \leq \sqrt{451 d^{2} + 1029.0519930 d + 587} \varepsilon \\ & \leq (21.2367606 d + 24.2280829) \varepsilon . \end{aligned} $$
Therefore, one can see that the result of this paper is better than that of the previous paper [8].