Near-isometries of the unit sphere
UDC 517.5We approximate $\varepsilon$-isometries of the unit sphere in $\ell_2^n$ and $\ell_\infty^n$ by linear isometries.
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2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512256052166656 |
|---|---|
| author | Vestfrid, I. A. Vestfrid, И. А. Vestfrid, І. А. |
| author_facet | Vestfrid, I. A. Vestfrid, И. А. Vestfrid, І. А. |
| author_sort | Vestfrid, I. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:01:38Z |
| description | UDC 517.5We approximate $\varepsilon$-isometries of the unit sphere in $\ell_2^n$ and $\ell_\infty^n$ by linear isometries. |
| doi_str_mv | 10.37863/umzh.v72i4.6049 |
| first_indexed | 2026-03-24T03:25:53Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
DOI: 10.37863/umzh.v72i4.6049
UDC 517.5
I. A. Vestfrid (Israel Inst. Technology, Haifa)
NEAR-ISOMETRIES OF THE UNIT SPHERE
БЛИЗЬКI IЗОМЕТРIЇ СФЕРИ ОБ’ЄДНАННЯ
We approximate \varepsilon -isometries of the unit sphere in \ell n2 and \ell n\infty by linear isometries.
Наведено наближення \varepsilon -iзометрiй одиничної сфери в \ell n2 i \ell n\infty лiнiйними iзометрiями.
1. Introduction. Notation. Throughout the paper X and Y denote real normed spaces. The
sphere and closed ball with center z and radius r are denoted by S(z, r) and B(z, r); we also write
S(0, r) = S(r) and B(0, r) = B(r). The unit sphere and ball are denoted by S and B (or SE and
BE when we need to specify the space). For a point x in \BbbR n, xi denotes its ith coordinate in the
standard basis \{ ei\} ni=1.
A local version of the classical Mazur – Ulam theorem asserts that a local isometry f, which maps
an open connected subset of X onto an open subset of Y, is the restriction of an affine isometry of
X onto Y (see, for example, [1, p. 341]). This classical result was generalized in several directions.
One of them is the study of the isometric extension problem posed by D. Tingley [8]: Let T be
a surjective isometry between the spheres of X and Y. Is T necessarily the restriction of a linear
isometry between X and Y ? There are a number of publications devoted to Tingley’s problem (see
[2] for a survey of corresponding results) and, in particular, the problem is solved in positive for
many concrete classical Banach spaces.
When distances are known only imprecisely, it is natural to study how close f is to be an
isometry. There are various different useful concepts of an approximate isometry, and one may then
ask whether such a mapping, which only nearly preserves distances, can be well approximated by a
true isometry, especially by an affine isometry (see [1], Chapters 14 and 15, and surveys [6] and [7]
for more complete exposition and literature on this subject).
Definition. Let A be a subset of X and \varepsilon \geq 0. A map f : A \rightarrow Y is called an \varepsilon -isometry if\bigm| \bigm| \| f(x) - f(y)\| - \| x - y\|
\bigm| \bigm| \leq \varepsilon (1)
for all x, y \in A.
The author [9, 11] has presented sharp results on approximation of \varepsilon -isometries of balls in \ell n2
and \ell n\infty .
In the present paper we study approximation of \varepsilon -isometries of spheres in \ell n2 and \ell n\infty . We give
the following results, proceeding the way of [9, 11].
c\bigcirc I. A. VESTFRID, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4 575
576 I. A. VESTFRID
Theorem 1. Let f : S\ell n2
\rightarrow S\ell n2
be an \varepsilon -isometry. Then there is a linear isometry U of \ell n2 such
that
\| f(x) - Ux\| \leq C \mathrm{l}\mathrm{o}\mathrm{g}(n+ 1)\varepsilon , x \in S\ell n2
, (2)
for some absolute constant C.
The upper bound in (2) is sharp.
Theorem 2. There are absolute constants C and c with the following property: Let 0 < \varepsilon < c
and f : S\ell n\infty \rightarrow S\ell n\infty be an \varepsilon -isometry. Then there is a unique linear isometry U of \ell n\infty such that
\| f(x) - Ux\| \leq C\varepsilon , x \in S\ell n\infty . (3)
2. Proofs. We need the following lemma, which is proven, in fact, by the proof of Lemma 6
in [9].
Lemma 1. Let A be a subset of X and f : A \rightarrow Y be an \varepsilon -isometry. Then there is a continuous
5\varepsilon -isometry f1 : A \rightarrow Y such that \| f1(x) - f(x)\| \leq 2\varepsilon for every x \in A.
(The word ”open” is redundant in the statement of [9], Lemma 6.)
Proof of Theorem \bfone . By Lemma 1, we can assume that f is continuous. We will use the
following statement that follows from [10] (Theorem II).
Proposition 1. Let f : B\ell n2
\rightarrow B\ell n2
be a continuous map satisfying
| \langle f(x), f(y)\rangle - \langle x, y\rangle | \leq \varepsilon .
Then there are an absolute constant C and a linear isometry U such that
\| f(x) - Ux\| \leq C \mathrm{l}\mathrm{o}\mathrm{g}(n+ 1)\varepsilon , x \in B\ell n2
.
Define \~f : B \rightarrow B by
\~f(x) =
\left\{ 0, x = 0,
\| x\| f
\biggl(
x
\| x\|
\biggr)
, otherwise.
Then
2
\bigm| \bigm| \bigm| \langle \~f(x), \~f(y)\rangle - \langle x, y\rangle
\bigm| \bigm| \bigm| =
= 2\| x\| \| y\|
\bigm| \bigm| \bigm| \bigm| \biggl\langle f \biggl( x
\| x\|
\biggr)
, f
\biggl(
y
\| y\|
\biggr) \biggr\rangle
-
\biggl\langle
x
\| x\|
,
y
\| y\|
\biggr\rangle \bigm| \bigm| \bigm| \bigm| =
= \| x\| \| y\|
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl( \bigm\| \bigm\| \bigm\| \bigm\| f \biggl( x
\| x\|
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| 2 - 2
\biggl\langle
f
\biggl(
x
\| x\|
\biggr)
, f
\biggl(
y
\| y\|
\biggr) \biggr\rangle
+
\bigm\| \bigm\| \bigm\| \bigm\| f \biggl( y
\| y\|
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| 2
\Biggr)
-
-
\Biggl( \bigm\| \bigm\| \bigm\| \bigm\| x
\| x\|
\bigm\| \bigm\| \bigm\| \bigm\| 2 - 2
\biggl\langle
x
\| x\|
,
y
\| y\|
\biggr\rangle
+
\bigm\| \bigm\| \bigm\| \bigm\| y
\| y\|
\bigm\| \bigm\| \bigm\| \bigm\| 2
\Biggr) \bigm| \bigm| \bigm| \bigm| \bigm| =
= \| x\| \| y\|
\bigm| \bigm| \bigm| \bigm| \bigm|
\bigm\| \bigm\| \bigm\| \bigm\| f \biggl( x
\| x\|
\biggr)
- f
\biggl(
y
\| y\|
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| 2 - \bigm\| \bigm\| \bigm\| \bigm\| x
\| x\|
- y
\| y\|
\bigm\| \bigm\| \bigm\| \bigm\| 2
\bigm| \bigm| \bigm| \bigm| \bigm| =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
NEAR-ISOMETRIES OF THE UNIT SPHERE 577
= \| x\| \| y\|
\bigm| \bigm| \bigm| \bigm| \bigm\| \bigm\| \bigm\| \bigm\| f \biggl( x
\| x\|
\biggr)
- f
\biggl(
y
\| y\|
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| - \bigm\| \bigm\| \bigm\| \bigm\| x
\| x\|
- y
\| y\|
\bigm\| \bigm\| \bigm\| \bigm\| \bigm| \bigm| \bigm| \bigm| \times
\times
\bigm| \bigm| \bigm| \bigm| \bigm\| \bigm\| \bigm\| \bigm\| f \biggl( x
\| x\|
\biggr)
- f
\biggl(
y
\| y\|
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| + \bigm\| \bigm\| \bigm\| \bigm\| x
\| x\|
- y
\| y\|
\bigm\| \bigm\| \bigm\| \bigm\| \bigm| \bigm| \bigm| \bigm| \leq 4\varepsilon .
The result follows by Proposition 1.
Sharpness of the estimate in (2):
Let n be an even natural number, say n = 2m. Following Matouškova [4], identify \BbbR n with
\BbbC m. For z \in \BbbC set \varphi \varepsilon (z) = zei\varepsilon log | z| (if z = 0 then \varphi \varepsilon (z) = 0). Now set
f\varepsilon (z1, . . . , zm) = (\varphi \varepsilon (z1), . . . , \varphi \varepsilon (zm)).
Then f\varepsilon is an \varepsilon -isometry of B\ell n2
onto B\ell n2
and, for every 0 < t \leq 1, f\varepsilon (tS\ell n2
) = tS\ell n2
.
Kalton in the proof of [3] (Proposition 2.1) actually proved that if 0 < \varepsilon <
2\pi
\mathrm{l}\mathrm{o}\mathrm{g}m
then, for any
affine isometry U of \ell n2 , we have
\mathrm{m}\mathrm{a}\mathrm{x}
x\in S\ell n2
\| f\varepsilon (x) - U(x)\| \geq \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
1
4
\varepsilon \mathrm{l}\mathrm{o}\mathrm{g}m
\biggr)
.
Proof of Theorem 2. We set c = 1/210 and fix an \varepsilon -isometry f, satisfying the conditions
of Theorem 2.
Lemma 2. There is a permutation \pi of \{ 1, . . . , n\} such that, for every i \leq n,
| f\pi (i)(ei) - f\pi (i)( - ei)| \geq 2 - \varepsilon
and
| fk(ei) - fk( - ei)| \leq 1 + \varepsilon , k \not = \pi (i).
Proof. Set, for each i \leq n,
Ai := \{ j \leq n : | fj(ei) - fj( - ei)| > 1 + \varepsilon \} .
By (1), we have, for every i \leq n,
\| f(ei) - f( - ei)\| \geq 2 - \varepsilon > 1 + \varepsilon . (4)
Thus Ai is nonempty.
Now we show that Ai \cap Ak = \varnothing for i \not = k, which implies that all Ai are disjoint singletons.
Assume to the contrary that there is j \in Ai \cap Ak, i.e.,
| fj(ei) - fj( - ei)| > 1 + \varepsilon and | fj(ek) - fj( - ek)| > 1 + \varepsilon .
Let \theta i, \theta k \in \{ - 1, 1\} be such that
| fj(ei) - fj( - ei)| + | fj(ek) - fj( - ek)| = fj(\theta iei) - fj( - \theta iei) + fj(\theta kek) - fj( - \theta kek).
Then fj(\theta iei) - fj( - \theta iei) + fj(\theta kek) - fj( - \theta kek) > 2(1 + \varepsilon ).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
578 I. A. VESTFRID
On the other hand, since \| ei \pm ek\| = 1, we have \| f(\theta iei) - f(\theta kek)\| \leq 1 + \varepsilon for every
\theta i, \theta k \in \{ - 1, 1\} . In particular,
| fj(\theta iei) - fj( - \theta kek)| + | fj( - \theta iei) - fj(\theta kek)| \leq 2(1 + \varepsilon );
a contradiction.
It follows that if \{ j\} \not = Ai, then | fj(ei) - fj( - ei)| \leq 1 + \varepsilon by the definition of Ai; and if
\{ j\} = Ai, then | fj(ei) - fj( - ei)| \geq 2 - \varepsilon by (4). So, the desired permutation \pi is defined by
\pi (i) \in Ai.
Lemma 3. For every i \leq n, | f\pi (i)(\pm ei)| \geq 1 - \varepsilon , f\pi (i)(ei) and f\pi (i)( - ei) have opposite signs.
Proof. By the definition of \pi , we get
| f\pi (i)(ei)| \geq 2 - \varepsilon - | f\pi (i)( - ei)| \geq 1 - \varepsilon .
The final comment follows directly from | f\pi (i)(ei) - f\pi (i)( - ei)| \geq 2 - \varepsilon and | f\pi (i)(\pm ei)| \leq 1.
It follows that there is a map s : \{ 1, . . . , n\} \rightarrow \{ - 1, 1\} such that
\mathrm{s}\mathrm{g}\mathrm{n}f\pi (i)(ei) = s(i) and \mathrm{s}\mathrm{g}\mathrm{n}f\pi (i)( - ei) = - s(i), i \leq n. (5)
Denote by Hi the hyperplane \{ x : xi = 1\} and by H - i the hyperplane \{ x : xi = - 1\} . Denote
by Si and S - i the following (n - 1)-dimensional faces Si = S \cap Hi, S - i = S \cap H - i.
Lemma 4. For every i \leq n, f(ei) \in Ss(i)\pi (i) and f( - ei) \in S - s(i)\pi (i). Moreover, for any k \not = i
f(ei), f( - ei) /\in S\pm \pi (k).
Proof. Let f(ei) \in Sj and f( - ei) \in Sk for some - n \leq j, k \leq n.
Assume to the contrary that j \not = \pm \pi (i). Then j = s\pi (l) for some s = \pm 1 and l \not = i. Assume
without loss of generality s = s(l) = 1. By Lemma 3, fj( - el) \leq - 1+\varepsilon . Hence, | fj( - el) - fj(ei)| \geq
\geq 2 - \varepsilon , while \| el + ei\| = 1 is a contradiction. Thus, j \in \{ - \pi (i), \pi (i)\} . Similarly, k \in
\in \{ - \pi (i), \pi (i)\} .
By (5), the result follows.
It follows that f\pi (i)(ei) = s(i) and f\pi (i)( - ei) = - s(i).
Lemma 5. Let x \in S\pm i. Then f(x) /\in S\mp s(i)\pi (i).
Proof. Assume without loss of generality x \in Si. Then \| x - ei\| \leq 1, \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}
\bigl(
f(ei), S - s(i)\pi (i)
\bigr)
=
= \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}
\bigl(
Ss(i)\pi (i), S - s(i)\pi (i)
\bigr)
= 2 and the result follows.
Lemma 6. For every i \leq n, \| f(ei) - s(i)e\pi (i)\| \leq \varepsilon and \| f( - ei) + s(i)e\pi (i)\| \leq \varepsilon .
Proof. We prove only the first estimation; the second one can be proved following the same path.
Assume to the contrary that \| f(ei) - s(i)e\pi (i)\| > \varepsilon . Then there is k \not = i such that | f\pi (k)(ei)| > \varepsilon .
It follows by Lemma 4 that either
| f\pi (k)(ei) - f\pi (k)(ek)| > 1 + \varepsilon or | f\pi (k)(ei) - f\pi (k)( - ek)| > 1 + \varepsilon .
But this contradicts \| ei - ek\| = 1.
Lemma 7. For every - n \leq i \leq n, there exists a linear isometry Ui : \ell n\infty \rightarrow \ell n\infty such that
Uiei = s(i)e\pi (i) and
\| f(x) - Uix\| \leq 100\varepsilon , x \in Si.
Proof. We will use the following statement (see [11], Proposition 2).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
NEAR-ISOMETRIES OF THE UNIT SPHERE 579
Proposition 2. Let 0 < \varepsilon < 1/6. Let f : B\ell n\infty \rightarrow B\ell n\infty be a continuous \varepsilon -isometry with f(0) = 0.
Then there is a unique linear isometry U of \ell n\infty such that
\| f(x) - Ufx\| \leq 2\varepsilon , x \in B(1 - 2\varepsilon ).
Assume without loss of generality i \geq 1. As \| f(ei) - s(i)e\pi (i)\| \leq \varepsilon , f (BHi(1 - 3\varepsilon )) \subset
\subset BHs(i)\pi (i)
(f(ei), 1 - 2\varepsilon ) \subset Ss(i)\pi (i). Denote by V the linear isometry of \ell n\infty defined by
V
\left( \sum
k/\in \{ i,\pi (i)\}
(xkek) + xiei + x\pi (i)e\pi (i)
\right) =
=
\sum
k/\in \{ i,s(i)\pi (i)\}
(xkek) + s(i)x\pi (i)ei + xie\pi (i).
Denote F1 = V \circ f. Then \| F1(ei) - ei\| \leq \varepsilon and F1 is an \varepsilon -isometry such that
F1 (BHi(1 - 3\varepsilon )) \subset BHi(F1(ei), 1 - 2\varepsilon ) \subset Si.
We consider now Hi as an (n - 1)-dimensional normed space with the origin \varnothing Hi = ei. Then
BHi = Si and \| F1(0)\| \leq \varepsilon .
Define a map F2 : BHi \rightarrow BHi by F2(x) = F1((1 - 3\varepsilon )x) - F1(0). Then F2 is an 7\varepsilon -isometry
with F2(0) = 0. By Lemma 1, there is a continuous 35\varepsilon -isometry F3 : BHi \rightarrow BHi such that
\| F3(x) - F2(x)\| \leq 14\varepsilon for every x \in BHi . By the choice of \varepsilon , F3 satisfies the conditions of
Proposition 2. Hence there is a linear (in Hi) isometry U such that
\| F3(x) - Ux\| \leq 70\varepsilon , x \in BHi(1 - 2\varepsilon ).
Since F1(x) = F2
\biggl(
x
1 - 3\varepsilon
\biggr)
+ F1(0), we have, on BHi(1 - 5\varepsilon ),
\| F1(x) - Ux\| \leq
\leq
\bigm\| \bigm\| \bigm\| \bigm\| F2
\biggl(
x
1 - 3\varepsilon
\biggr)
- F3
\biggl(
x
1 - 3\varepsilon
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| + \bigm\| \bigm\| \bigm\| \bigm\| F3
\biggl(
x
1 - 3\varepsilon
\biggr)
- U
\biggl(
x
1 - 3\varepsilon
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| +
+
3\varepsilon
1 - 3\varepsilon
\| Ux\| + \| F1(0)\| < 89\varepsilon .
Let x \in BHi \setminus BHi(1 - 5\varepsilon ). Then
\| F1(x) - Ux\| \leq
\leq
\bigm\| \bigm\| \bigm\| \bigm\| F1(x) - F1
\biggl(
(1 - 5\varepsilon )x
\| x\|
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| + \bigm\| \bigm\| \bigm\| \bigm\| Ux - U
(1 - 5\varepsilon )x
\| x\|
\bigm\| \bigm\| \bigm\| \bigm\| +
+
\bigm\| \bigm\| \bigm\| \bigm\| F1
\biggl(
(1 - 5\varepsilon )x
\| x\|
\biggr)
- U
(1 - 5\varepsilon )x
\| x\|
\bigm\| \bigm\| \bigm\| \bigm\| <
< 2(\| x\| - 1 + 5\varepsilon ) + 90\varepsilon \leq 100\varepsilon .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
580 I. A. VESTFRID
Obviously, U is the restriction on Hi of the linear isometry U \prime of \ell n\infty defined by
U \prime
\left( \sum
k \not =i
(xkek) + xiei
\right) =
\sum
k \not =i
U(xkek) + xiei.
Thus, Ui = V - 1U \prime is a desired isometry.
Define a linear isometry U by
U
\Biggl( \sum
i
xiei
\Biggr)
=
\sum
i
s(i)xie\pi (i).
Lemma 8. For every - n \leq i \leq n, Ui = U.
Proof. Suppose that Ui \not = U for some i. Then there is j \not = | i| such that Uiej \not = Uej . Assume
without loss of generality i, j \geq 1. Then there are k /\in \{ \pi (i), \pi (j)\} and s \in \{ - 1, 1\} such that
Uiej = sek. Hence,
\| Ui(ei + ej) - Uj(ei + ej)\| = \| s(i)e\pi (i) + sek - Ujei - s(j)e\pi (j)\| \geq 1.
On the other hand, by Lemma 7
\| Ui(ei + ej) - Uj(ei + ej)\| \leq
\leq \| Ui(ei + ej) - f(ei + ej)\| + \| f(ei + ej) - Uj(ei + ej)\| \leq 200\varepsilon < 1;
a contradiction.
Thus, U satisfies (3) with C = 100.
Uniqueness: Suppose that U \prime is another linear isometry of \ell n\infty satisfying (3) with C = 100.
Then
\| f(ei) - Uei\| \leq 100\varepsilon and \| f(ei) - U \prime ei\| \leq 100\varepsilon for every i \leq n.
Thus, \| U \prime ei - Uei\| \leq 200\varepsilon < 1, which means U \prime ei = Uei.
References
1. Y. Benyamini, J. Lindenstrauss, Geometric nonlinear functional analysis, vol. 1, Amer. Math. Soc., Providence, RI
(2000).
2. G. Ding, On isometric extension problem between two unit spheres, Sci. China Ser. A, 52, 2069 – 2083 (2009).
3. N. J. Kalton, A remark on quasi-isometries, Proc. Amer. Math. Soc., 131, 1225 – 1231 (2003).
4. E. Matouškova, Almost isometries of balls, J. Funct. Anal., 190, 507 – 525 (2002).
5. S. Mazur, S. Ulam, Sur les transformations isométriques d’espaces vectoriels normés, Comp. Rend. Paris, 194,
946 – 948 (1932).
6. Th. M. Rassias, Properties of isometries and approximate isometries. Recent progress in inequalities (Niš, 1996),
Math. Appl., 430, 341 – 379 (1998).
7. Th. M. Rassias, Properties of isometric mappings, J. Math. Anal. and Appl., 235, 108 – 121 (1999).
8. D. Tingley, Isometries of the unit sphere, Geom. Dedicata, 22, 371 – 378 (1987).
9. I. A. Vestfrid, \varepsilon -Isometries in Euclidean spaces, Nonlinear Anal., 63, 1191 – 1198 (2005).
10. I. A. Vestfrid, Addendum to \varepsilon -isometries in Euclidean spaces, Nonlinear Anal., 63, 1191–1198 (2005); 67, 1306–1307
(2007).
11. I. A. Vestfrid, \varepsilon -Isometries in ln\infty , Nonlinear Funct. Anal. and Appl., 12, 433 – 438 (2007).
Received 20.03.18
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
|
| id | umjimathkievua-article-6049 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:53Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2c/fe02928b407c394dad84b07868cb0a2c.pdf |
| spelling | umjimathkievua-article-60492022-03-26T11:01:38Z Near-isometries of the unit sphere Near-isometries of the unit sphere Vestfrid, I. A. Vestfrid, И. А. Vestfrid, І. А. UDC 517.5We approximate $\varepsilon$-isometries of the unit sphere in $\ell_2^n$ and $\ell_\infty^n$ by linear isometries. УДК 517.5 Наведено наближення $\varepsilon$-ізометрій одиничної сфери в $\ell_2^n$ і $\ell_\infty^n$ лінійними ізометріями. Institute of Mathematics, NAS of Ukraine 2020-03-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6049 10.37863/umzh.v72i4.6049 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 4 (2020); 575-580 Український математичний журнал; Том 72 № 4 (2020); 575-580 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6049/8709 |
| spellingShingle | Vestfrid, I. A. Vestfrid, И. А. Vestfrid, І. А. Near-isometries of the unit sphere |
| title | Near-isometries of the unit sphere |
| title_alt | Near-isometries of the unit sphere |
| title_full | Near-isometries of the unit sphere |
| title_fullStr | Near-isometries of the unit sphere |
| title_full_unstemmed | Near-isometries of the unit sphere |
| title_short | Near-isometries of the unit sphere |
| title_sort | near-isometries of the unit sphere |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6049 |
| work_keys_str_mv | AT vestfridia nearisometriesoftheunitsphere AT vestfridia nearisometriesoftheunitsphere AT vestfridía nearisometriesoftheunitsphere |