Paper-6-Acta23.2010. 109KB Jun 04 2011 12:03:14 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 23/2010
pp. 69-77

ZARANTONELLO’S INEQUALITY AND THE ISOMETRIES OF
THE M-DIMENSIONAL EUCLIDEAN SPACE

Teodora Andrica
Abstract. The main purpose of this paper is to use Zarantonello’s inequality
(see reference [9]) in order to prove that any isometry of the Euclidean m-space is
affine, and then describe all the isometries of the Euclidean m-space.
2000 Mathematics Subject Classification: 51B20; 51F99; 51K05; 51K99; 51N25.
Keywords and phrases: isometry with respect to a metric, group of isometries,
translations group of the Euclidean m-space, Mazur-Ulam theorem.
1. Introduction and preliminaries
Let (X, d) be a metric space. The map f : X → X is called an isometry
with respect to the metric d (or a d-isometry), if f is surjective and preserves the
distances. That is for any points x, y ∈ X the relation d(f (x), f (y)) = d(x, y)
holds. From this relation it follows that the map f is injective, hence it is bijective.

Denote by Isod (X) the set of all isometries of the metric space (X, d). It is clear that
(Isod (X), ◦) is a subgroup of (S(X), ◦), where S(X) denotes the group of all bijective
transformations f : X → X. We will call (Isod (X), ◦) the group of isometries of the
metric space (X, d). A general, important and complicated problem is to describe
the group (Isod (X), ◦). This problem was formulated in paper [1] for metric spaces
with a metric that is not given by a norm and it is of great interest (see for instance
[3]).
In this article we want to prove a standard result of Ulam concerning the group of
isometries of the Euclidean space Rm , using the so-called Zarantonello’s inequality.
In the space Rm we consider the Euclidean metric, which is defined by the inner
product. The inner product of two vectors x, y ∈ Rm is given by
hx, yi = x1 y 1 + x2 y 2 + · · · + xm y m .
The norm of vector x is given by
p
p
kxk = hx, xi = (x1 )2 + (x2 )2 + · · · + (xm )2 .
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T. Andrica - Zarantonello’s inequality and the isometries of the m-dimensional...

The Euclidean metric is defined by
p
d(x, y) = kx − yk = (x1 − y 1 )2 + (x2 − y 2 )2 + · · · + (xm − y m )2 .
Here are few standard examples of isometries with respect to this metric.
Example 1.
1. If A ∈ O(m) is an orthogonal matrix, i.e. AT A = I, then the
linear map fA : Rm → Rm , defined by fA (x) = Ax is an isometry.
2. If b ∈ Rm , then the translation of vector b, tb : Rm → Rm , tb (x) = x + b, is an
isometry. Moreover, the inverse of tb is t−b .
From the well-known result of S.Mazur and S.Ulam (see the original reference
[5]) we have:
Every isometry f : E → F between real normed spaces is affine. In this case an
isometry is a surjective map satisfying for any x, y ∈ E the relation ||f (x)−f (y)||F =
||x − y||E . This result was proved by S.Mazur and S.Ulam in 1932. A simple proof
was given by J.Vaisala [7], the proof is based on the ideas of A.Vogt [8], and it uses
reflections in points. If we apply this result for the normed spaces E = F = Rm
with the Euclidean norm k · k, then it follows that any isometry is affine. Therefore,
it is sufficient to see which are the affine maps that preserve the distances.
Our purpose is to use Zarantonello’s inequality (see the reference [9]) in order
to prove that any isometry of the Euclidean m-space is affine, and then describe all

the isometries of the Euclidean m-space. In paper [2] we have used the same idea in
the complex plane. In this respect we need few auxiliary results.
Theorem 1 (Lagrange’s theorem). Let n, m be positive integers, let x1 , x2 , . . . , xn ∈
Rm and α1 , α2 , . . . , αn ∈ R be real numbers such that α1 + · · · + αn = 1. Then for
each x ∈ Rm the following relation holds:
n
X

2

2

αk kx − xk k = kx − α1 x1 − · · · − αn xn k +

k=1

n
X

αk kxk − α1 x1 − · · · − αn xn k2 .

k=1

Proof. Using the properties of the inner product, we get:
n
X
k=1

αk kx − xk k2 =

n
X

αk hx − xk , x − xk i =

k=1

= kxk2 − 2

n

X

αk (kxk2 − 2hx, xk i + kxk k2 )

k=1

n
X

αk hx, xk i +

k=1

n
X
k=1

70

αk kxk k2 .

(1)

T. Andrica - Zarantonello’s inequality and the isometries of the m-dimensional...

On the other hand:
kx − α1 x1 − · · · − αn xn k2 +

n
X

αk kxk − α1 x1 − · · · − αn xn k2

k=1

= hx − α1 x1 − · · · − αn xn , x − α1 x1 − · · · − αn xn i
+

n
X

αk hxk − α1 x1 − · · · − αn xn , xk − α1 x1 − · · · − αn xn i

k=1
2

= kxk − 2

n
X

2

αk hx, xk i + 2kα1 x1 + · · · + αn xn k +

k=1

−2

n

X

n
X

αk kxk k2

k=1

2

αk hxk , α1 x1 + · · · + αn xn i = kxk − 2

k=1

n
X

αk hx, xk i +

k=1

n
X

αk kxk k2 .

(2)

k=1

From (1) and (2) we obtain the desired relation.
Theorem 2 (Stewart’s theorem). Let m be a positive integer, x1 , x2 ∈ Rm and
a ∈ R. Then for each x ∈ Rm the following relation holds:
kx − ax1 − (1 − a)x2 k2 = akx − x1 k2 + (1 − a)kx − x2 k2 − a(1 − a)kx1 − x2 k2 .
Proof. Using the properties of the inner product, the left hand side of the desired
relation can be written as:
kx − ax1 − (1 − a)x2 k2 = kx + ax − ax − ax1 − (1 − a)x2 k2
= ka(x − x1 ) + (1 − a)(x − x2 )k2
= ha(x − x1 ) + (1 − a)(x − x2 ), a(x − x1 ) + (1 − a)(x − x2 )i

= a2 kx − x1 k2 + (1 − a)2 kx − x2 k2
+2a(1 − a)hx − x1 , x − x2 i = a2 kx − x1 k2 + (1 − a)2 kx − x2 k2
+2a(1 − a)(kxk2 − hx, x1 i − hx, x2 i + hx1 , x2 i).
We get:
kx−ax1 −(1−a)x2 k2 = a2 kx−x1 k2 +(1−a)2 kx−x2 k2 +a(1−a)(kx−x1 k2 +kx−x2 k2
−kx1 k2 − kx2 k2 + 2hx1 , x2 i) = akx − x1 k2 + (1 − a)kx − x2 k2 − a(1 − a)kx1 − x2 k2 .

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T. Andrica - Zarantonello’s inequality and the isometries of the m-dimensional...

Stewart’s theorem can be extended in the following way:
Theorem 3. Let n, m be positive integers, n ≥ 2, m ≥ 1, x1 , x2 , . . . , xn ∈ Rm ,
a1 , a2 , . . . , an ∈ R such that a1 + · · · + an = 1. Then for each x ∈ Rm the following
relation holds:
2

n
n

X
X
X


ak xk
=
ak kx − xk k2 −
ak al kxk − xl k2 .
x −


k=1

k=1

1≤k