Betweenness relation and isometric imbeddings of metric spaces
We give an elementary proof of the classical Menger result according to which any metric space X that consists of more than four points is isometrically imbedded into \( \mathbb{R} \) if every three-point subspace of X is isometrically imbedded into \( \mathbb{R} \). A series of corollaries of this...
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| Date: | 2009 |
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Institute of Mathematics, NAS of Ukraine
2009
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| author | Dovgoshei, A. A. Dordovskii, D. V. Довгошей, А. А. Дордовский, Д. В. Довгошей, А. А. Дордовский, Д. В. |
| author_facet | Dovgoshei, A. A. Dordovskii, D. V. Довгошей, А. А. Дордовский, Д. В. Довгошей, А. А. Дордовский, Д. В. |
| author_sort | Dovgoshei, A. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:45:28Z |
| description | We give an elementary proof of the classical Menger result according to which any metric space X that consists of more than four points is isometrically imbedded into \( \mathbb{R} \) if every three-point subspace of X is isometrically imbedded into \( \mathbb{R} \). A series of corollaries of this theorem is obtained. We establish new criteria for finite metric spaces to be isometrically imbedded into \( \mathbb{R} \). |
| first_indexed | 2026-03-24T02:36:17Z |
| format | Article |
| fulltext |
UDK 515.124.4
A. A. Dovhoßej, D. V. Dordovskyj
(In-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck)
OTNOÍENYE LEÛAT| MEÛDU Y YZOMETRYÇESKYE
VLOÛENYQ METRYÇESKYX PROSTRANSTV
We give an elementary proof of the classical Menger result according to which every metric space X
with cardinality more than four can be isometrically embedded in R if all three-point subspaces of X
have the same embeddings in R. A number of corollaries of this theorem are obtained. New criteria for
the ability of finite metric spaces to be isometrically embedded in R are found.
Navedeno elementarne dovedennq klasyçnoho rezul\tatu K. Menhera pro te, wo bud\-qkyj met-
ryçnyj prostir X, wo sklada[t\sq bil\ß niΩ z çotyr\ox toçok, izometryçno vklada[t\sq v R,
qkwo koΩnyj trytoçkovyj pidprostir X izometryçno vklada[t\sq v R. Otrymano rqd naslid-
kiv z ci[] teoremy. Vstanovleno novi kryteri] izometryçno] vkladenosti v R skinçennyx metryç-
nyx prostoriv.
1. Vvedenye y formulyrovka rezul\tatov. Ternarnoe otnoßenye „leΩat\
meΩdu” v qvnoj forme vperv¥e b¥lo opredeleno D. Hyl\bertom v eho znameny-
t¥x „Osnovanyqx heometryy” [1]. Analohyçn¥e otnoßenyq estestvenn¥m obra-
zom voznykagt v razlyçn¥x oblastqx matematyky: teoryy çastyçno uporqdo-
çenn¥x mnoΩestv, lynejn¥x prostranstv, uporqdoçenn¥x polej y t. d. (sm., na-
prymer, [2, 3]). V teoryy metryçeskyx prostranstv ponqtye „metric betweenness”,
ysxodnoe dlq nastoqwej rabot¥, b¥lo vvedeno K. Menherom [4] v sledugwej
forme.
Opredelenye 1.1. Pust\ ( X, d ) — metryçeskoe prostranstvo, a x, y, z —
razlyçn¥e toçky X. Budem hovoryt\, çto y leΩyt meΩdu x y z , esly
d ( x, z ) = d ( x, y ) + d ( y, z ).
Tak opredelennoe otnoßenye „leΩat\ meΩdu” qvlqetsq fundamental\n¥m v
teoryy heodezyçeskyx na metryçeskyx prostranstvax (sm., naprymer, [5]) y es-
testvenn¥m obrazom voznykaet pry yzuçenyy nayluçßyx pryblyΩenyj v metry-
çeskyx prostranstvax [6].
Xarakterystyçeskye svojstva ternarn¥x otnoßenyj, qvlqgwyxsq otnoße-
nyem „metric betweenness” dlq (dejstvytel\noznaçn¥x) metryk b¥ly najden¥
A.<Val\dom [7]. V dal\nejßem problem¥ metryzacyy otnoßenyq „leΩat\
meΩdu” (ne obqzatel\no dejstvytel\noznaçn¥my metrykamy) rassmatryvalys\ v
rabotax [8 – 10]. Nedavno dlq „metric betweenness” b¥ly najden¥ analohy klas-
syçeskyx teorem Syl\vestra – Hallay y Brajna – ∏rdeßa [11 – 13].
Zameçanye 1.1. Lehko proveryt\, çto dlq trex razlyçn¥x toçek x, y, z ∈ X
ravenstvo
2 max { d ( x, y ) , d ( x, z ), d ( y, z ) } = d ( x, y ) + d ( x, z ) + d ( y, z )
v¥polnqetsq tohda y tol\ko tohda, kohda odna yz πtyx toçek leΩyt meΩdu dvu-
mq druhymy. Druhym neobxodym¥m y dostatoçn¥m uslovyem qvlqetsq ravenstvo
nulg opredelytelq Kπly – Menhera (sm., naprymer, [14, s. 290])
det
, ,
, ,
,
0 1
0 1
2 2
2 2
2
d x y d x z
d y x d y z
d z x d
( ) ( )
( ) ( )
( ) 22 0 1
1 1 1 0
( )z y,
= 0.
© A. A. DOVHOÍEJ, D. V. DORDOVSKYJ, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10 1319
1320 A. A. DOVHOÍEJ, D. V. DORDOVSKYJ
Naybolee sloΩnaq çast\ rabot¥ K. Menhera [4] svqzana s dokazatel\stvom
toho, çto metryçeskoe prostranstvo X s card X ≥ n + 4 yzometryçno vklad¥-
vaetsq v evklydovo prostranstvo En
, esly lgboe A ⊆ X s card A = n + 2 ymeet
πto svojstvo. Teorema 1.1, sformulyrovannaq nyΩe, po suty πkvyvalentna pry-
vedennomu rezul\tatu K. Menhera dlq sluçaq n = 1 (πlementarnoe dokazatel\-
stvo πtoj teorem¥ dano vo vtoroj çasty nastoqwej stat\y).
Opredelenye 1.2. Metryçeskoe prostranstvo ( X, d ) ymeet M -svojst-
vo, esly sredy lgb¥x trex eho razlyçn¥x toçek najdetsq odna, leΩawaq meΩdu
dvumq druhymy.
Çerez � budem oboznaçat\ klass vsex metryçeskyx prostranstv, ymegwyx
M-svojstvo.
Esly prostranstvo ( X, d ) yzometryçno vklad¥vaetsq v ( R, | ⋅, ⋅ | ), to ono,
oçevydno, ymeet M-svojstvo, no obratnoe, voobwe hovorq, neverno.
Prymer 1.1. Pust\ t y s — poloΩytel\n¥e dejstvytel\n¥e çysla. Oboz-
naçym çerez A t s4( ), çet¥rextoçeçnoe metryçeskoe prostranstvo { a, b, c, f },
rasstoqnyq meΩdu toçkamy kotoroho udovletvorqgt ravenstvam
d ( a, b ) = d ( c, f ) = s, d ( b, c ) = d ( f, a ) = t, d ( a, c ) = d ( b, f ) = s + t.
Lehko proveryt\, çto dannoe prostranstvo ymeet M-svojstvo. Odnako A t s4( ),
ne vklad¥vaetsq v R, tak kak diam ( A4 ( t, s ) ) = d ( a, c ) = d ( b, f ), hde vse çet¥re
toçky a, b, c, f poparno razlyçn¥, çto nevozmoΩno dlq podmnoΩestva R.
Teorema 1.1. Pust\ X — metryçeskoe prostranstvo, ymegwee M-svoj-
stvo. Tohda lybo najdutsq t, s > 0 takye, çto X yzometryçno A t s4( ), , ly-
bo X yzometryçno nekotoromu podmnoΩestvu R.
Sledstvye 1.1. Pust\ X ∈ � y card X ≥ 5, tohda X yzometryçno vkla-
d¥vaetsq v R .
Sledstvye 1.2. Pust\ X ∈ �, tohda dlq lgboj toçky a ∈ X najdetsq
okrestnost\ U � a, yzometryçno vloΩymaq v R.
Teorema 1.1 pozvolqet takΩe utverΩdat\ nalyçye v � „maksymal\n¥x po
vloΩenyg” πlementov.
Opredelenye 1.3. Budem hovoryt\, çto prostranstvo X ∈ � maksymal\-
no po vloΩenyg, esly lgboe yzometryçeskoe vloΩenye f : X → Y s Y ∈ � qv-
lqetsq yzometryej.
Sledstvye 1.3. Prostranstva A t s4( ), y R maksymal\n¥ po vloΩenyg
v klasse � y lgboj πlement X ∈ � yzometryçno vloΩym lybo v R , lybo v
A t s4( ), dlq nekotor¥x t, s > 0.
Dokazatel\stvo. Proverym maksymal\nost\ R (ostal\noe lehko sleduet
yz teorem¥ 1.1). Pust\ f : R → X, X ∈ �, — yzometryçnoe vloΩenye takoe, çto
X \ f ( R ) ≠ ∅.
V¥berem a ∈ X \ f ( R ) y b ∈ f ( R ). Pust\ s : = d ( a, b ), hde d — metryka na X.
Tohda suwestvugt dve razlyçn¥e toçky p1
, p2 ∈ R , dlq kotor¥x
f b p f b p− −( ) − = ( ) −1
1
1
2 = s.
Sledovatel\no, b f p− ( )1 = b f p− ( )2 = b a− = s > 0. V sylu sledstvyq 1.1
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
OTNOÍENYE LEÛAT| MEÛDU Y YZOMETRYÇESKYE VLOÛENYQ … 1321
X yzometryçno vloΩymo v R. Pust\ g : X → R — takoe vloΩenye. Tohda toç-
ky g ( a ), g ( f ( p1 ) ), g ( f ( p2 ) ) razlyçn¥ y leΩat na rasstoqnyy s > 0 ot toçky
g ( b ). Poslednee nevozmoΩno, tak kak lgbaq „sfera” v R sostoyt rovno yz
dvux toçek.
Sledstvye dokazano.
Pered tem kak perejty k sledugwemu sledstvyg napomnym, çto lynejnoe
normyrovannoe prostranstvo ( X, || ⋅ || ) budet stroho v¥pukl¥m tohda y tol\ko
tohda, kohda ravenstvo
|| x + y || = || x || + || y ||, x, y ∈ X,
vleçet lynejnug zavysymost\ x y y [15, s. 496].
Sledstvye 1.4. Pust\ X — metryçeskoe prostranstvo, ymegwee M -
svojstvo, a ( Y, ⋅ ) — stroho v¥pukloe, normyrovannoe prostranstvo s
card Y > 1. Tohda lybo X yzometryçno vloΩymo v Y , lybo najdutsq t, s > 0
takye, çto X yzometryçno A t s4( ), .
Dokazatel\stvo. Dostatoçno pokazat\, çto yz vloΩymosty X v Y sledu-
et vloΩymost\ X v R. Rassmotrym vnaçale sluçaj, kohda Y — lynejnoe pros-
transtvo nad polem dejstvytel\n¥x çysel. Esly f : X → Y — yzometryq y p ∈
∈ X, to otobraΩenye
g : X → Y, g ( x ) : = f ( x ) – f ( p )
toΩe qvlqetsq yzometryej, dlq kotoroj g ( p ) = 0. Esly x1 ∈ X y x1 ≠ p, to
pryvedenn¥j v¥ße kryteryj strohoj v¥puklosty pokaz¥vaet, çto dlq lgboho
x ∈ X suwestvuet edynstvennoe λ = λ ( x ) ∈ R takoe, çto g ( x ) = λ g ( x1 ). Ne-
trudno proveryt\, çto otobraΩenye
X � x � λ || g ( x1 ) || ∈ R
qvlqetsq yskom¥m yzometryçeskym vloΩenyem v R.
Esly Y — lynejnoe prostranstvo nad C , to, rassuΩdaq analohyçno, polu-
çaem yzometryçeskoe vloΩenye X v C. Ostalos\ zametyt\, çto kompleksnaq
ploskost\ C est\ stroho v¥pukloe prostranstvo nad R.
Sledstvye dokazano.
Sledugwye prymer¥ pokaz¥vagt, çto uslovye strohoj v¥puklosty prost-
ranstva Y, voobwe hovorq, ne moΩet b¥t\ opuweno v sledstvyy 1.4.
Prymer 1.2. Rassmotrym prostranstvo l2
∞
dvuçlenn¥x çyslov¥x posledo-
vatel\nostej x = ( x1 , x2 ) s normoj || x || ∞ = max { | x1 |, | x2 | }. Tohda lehko pro-
veryt\, çto eho çet¥rextoçeçnoe podmnoΩestvo
{ ( 0, 0 ), ( s, s ), ( t, – t ), ( s + t, s – t ) }
s ynducyrovannoj yz l2
∞
metrykoj yzometryçno A t s4( ), .
Prymer 1.3. Druhoj model\g prostranstva A t s4( ), qvlqetsq çet¥rexto-
çeçnoe podmnoΩestvo { ( 0, 0 ), ( 0, t ), ( s, 0 ), ( s, t ) } prostranstva l2
1
s normoj
|| x || 1 = | x1 | + | x2 |.
Uslovyq ravenstva v neravenstve Mynkovskoho (sm., naprymer, [16, s. 42]) po-
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1322 A. A. DOVHOÍEJ, D. V. DORDOVSKYJ
kaz¥vagt, çto prostranstva ln
p
qvlqgtsq stroho v¥pukl¥my pry 1 < p < ∞.
Otsgda, yz sledstvyq 1.4 y prymerov 1.2, 1.3 poluçaem takoe sledstvye.
Sledstvye 1.5. Pust\ n — natural\noe çyslo ≥ 2 yly n = card N .
Tohda sledugwye utverΩdenyq πkvyvalentn¥:
i) lgboe X ∈ � vloΩymo v ln
p
;
ii) p = 1 yly p = ∞.
Zameçanye 1.2. Sovpadenye v¥puklosty s „metryçeskoj v¥puklost\g” qv-
lqetsq xarakterystyçeskym svojstvom stroho v¥pukl¥x lynejn¥x normyrovan-
n¥x prostranstv (sm. [17, 18]).
2. Dokazatel\stva y vspomohatel\n¥e utverΩdenyq. Vnaçale pryvedem
texnyçeskug lemmu.
Lemma 2.1. Pust\ A = { x1 , x2 , x3 , x4 } — çet¥rextoçeçnoe metryçeskoe
prostranstvo s metrykoj d, ymegwej M-svojstvo, y takoe, çto
diam A = d ( x1 , x2 ) > d ( x3 , x4 ). (2.1)
Tohda
d ( x3 , x4 ) < max { d ( x2 , x3 ), d ( x2 , x4 ) }. (2.2)
Rys. 1
Dokazatel\stvo. PoloΩym r : = d ( x1 , x2 ), s : = d ( x2 , x3 ), t : = d ( x3 , x1 ), r1
: = : = d ( x4 , x3 ), s1 : = d ( x4 , x1 ), t1 : = d ( x4 , x2 ) (sm. rys. 1, na kotorom verßyn¥
hrafa predstavlqgt soboj toçky metryçeskoho prostranstva { x1 , x2 , x3 , x4 }, a
vesa nad rebramy — rasstoqnyq meΩdu toçkamy).
PredpoloΩym, çto (2.2) ne v¥polnqetsq, tohda r1 ≥ max { t1 , s }. Yz posled-
neho neravenstva y yz (2.1), yspol\zuq M-svojstvo, v¥vodym
r = s + t, r = t1 + s1 , r1 = s + t1 . (2.3)
Krome toho, prymenqq M-svojstvo k treuhol\nyku s verßynamy x1 , x3 y x4 ,
poluçaem odno yz ravenstv
r1 = s1 + t, s1 = r1 + t, t = s1 + r1 . (2.4)
Esly r1 = s1 + t, to, yspol\zuq poslednee ravenstvo v (2.3), ymeem 2r1 = s1 +
+ t1 + t + s, a sklad¥vaq dva perv¥x ravenstva v (2.3), poluçaem 2r = s1 + t1 + t +
+ s. Otsgda r1 = r, çto protyvoreçyt (2.1). Analohyçno, yz vtoroho ravenstva
v (2.4) y posledneho v (2.3) ymeem s1 = s + t + t1 , a yz perv¥x dvux ravenstv yz
(2.3) naxodym s1 = s + t – t1 . Sledovatel\no, t1 = 0, çto nevozmoΩno, tak kak x4
≠ x2
. Nakonec, ravenstvo t = s1 + r1 y (2.3) pryvodqt k
s + s1 + t1 = t = – s + s1 + t1
.
çto protyvoreçyt uslovyg x2 ≠ x3
.
Lemma dokazana.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
OTNOÍENYE LEÛAT| MEÛDU Y YZOMETRYÇESKYE VLOÛENYQ … 1323
Napomnym, çto toçka p metryçeskoho prostranstva ( X, d ) naz¥vaetsq dya-
metral\noj dlq X, esly
sup ,
x X
d p x
∈
( ) = diam X
(sm., naprymer, [15, s. 497]). Analohyçno, budem hovoryt\, çto para { p, q } ⊆ X
qvlqetsq dyametral\noj, esly
d ( p, q ) = diam X.
Kak pokaz¥vaet sledugwee utverΩdenye, edynstvennost\ dyametral\noj pa-
r¥ y vloΩymost\ v R πkvyvalentn¥ dlq koneçn¥x metryçeskyx prostranstv
X ∈ �.
UtverΩdenye 2.1. Pust\ ( X, d ) — metryçeskoe prostranstvo, soderΩa-
wee koneçnoe çyslo toçek. Prostranstvo X yzometryçno vklad¥vaetsq v R
tohda y tol\ko tohda, kohda X ∈ � y dlq lgboho A ⊆ X s card A ≥ 2 su -
westvuet edynstvennaq para { a1 , a2 } ⊆ A, dlq kotoroj
diam A = d ( a1 , a2 ). (2.5)
Dokazatel\stvo. Esly X — koneçnoe podmnoΩestvo R, to edynstven-
nost\ dyametral\noj par¥ y M -svojstvo oçevydn¥. Proverym dostatoçnost\
πtyx uslovyj. Poskol\ku pry card X ≤ 3 vloΩymost\ πkvyvalentna M-svoj-
stvu, dalee predpolahaem n : = card X ≥ 4.
Zametym, çto yz lemm¥ 2.1, M-svojstva y edynstvennosty dyametral\noj
par¥ poluçaem sledugwee. Esly A ⊆ X, card A ≥ 3, a1 , a2 ∈ A y ymeet mesto
(2.5), to dlq lgboho B ⊆ A ravenstva 1 + card B = card A y diam B = d ( b1 , b2 )
vlekut
{ b1 , b2 } ∩ { a1 , a2 } ≠ ∅. (2.6)
Oboznaçym çerez x1 , xn toçky yz X, dlq kotor¥x
diam X = d ( x1 , xn ).
PoloΩym X1 : = X \ { xn }, tohda v sylu (2.6) y edynstvennosty dyametral\noj
par¥ ymeetsq edynstvennaq toçka xn – 1 ∈ X1 , dlq kotoroj diam X1 = d ( x1 ,
xn – 1 ). Analohyçno opredelqetsq xn – 2 kak edynstvennaq toçka yz X2 \ { xn ,
xn – 1 }, dlq kotoroj diam X2 = d ( x1 , xn – 2 ) y t. d. V rezul\tate poluçaem
numeracyg X = = { x1 , … , xn }.
Yskomoe vloΩenye f : X → R opredelym po pravylu f ( xk ) = d ( x1 , xk ) pry
1 ≤ k ≤ n. Tohda f ( xk ) = diam Xn – k ( pry k = n sçytaem X0 : = X ) y, v sylu
edynstvennosty dyametral\noj par¥, ymeem strohye neravenstva
f ( x1 ) < f ( x2 ) < … < f ( xn ).
Ostalos\ proveryt\, çto f soxranqet rasstoqnye, t. e.
| f ( xk ) – f ( xm ) | = d ( xk , xm ) (2.7)
dlq xk , xm ∈ X. Ne umen\ßaq obwnosty sçytaem 1 ≤ k < m ≤ n, tohda v sootvet-
stvyy s ynduktyvn¥my postroenyqmy ymeem x1 , xk , xm ∈ Xn – m y diam Xn m− =
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1324 A. A. DOVHOÍEJ, D. V. DORDOVSKYJ
= d ( x1 , xm ) ≥ max { d ( x1 , xk ), d ( xk , xm ) }. Sledovatel\no, xk leΩyt meΩdu x1 y
xm , t. e. f ( xm ) = d ( x1 , xm ) = d ( x1 , xk ) + d ( xk , xm ) = f ( xk ) + d ( xk , xm ) , otkuda po-
luçaem (2.7).
Sledstvye 2.1. Pust\ ( X, d ) — koneçnoe metryçeskoe prostranstvo,
ymegwee M -svojstvo. Tohda X ne vloΩymo v R v tom y tol\ko v tom slu-
çae, kohda najdetsq çet¥rextoçeçnoe mnoΩestvo A = { x1 , x2 , x3 , x4 } ⊆ X ta-
koe, çto
diam A = d ( x1 , x2 ) = d ( x3 , x4 ). (2.8)
Dokazatel\stvo. Esly X ne vloΩymo v R , to yz utverΩdenyq 2.1 sle-
duet, çto dlq nekotoroho B ⊆ X ymeem
d ( x1 , x2 ) = d ( x3 , x4 ) = diam B,
hde { x1 , x2 } y { x3 , x4 } — razlyçn¥e dvuxtoçeçn¥e podmnoΩestva B. Vsled-
stvye M-svojstva { x1 , x2 } y { x3 , x4 } ne peresekagtsq. Tohda A = { x1 , x2 , x3 ,
x4 } — çet¥rextoçeçnoe mnoΩestvo, dlq kotoroho ymeet mesto (2.8).
Sledstvye dokazano.
Zametym, çto pryvedennoe sledstvye y teorema 1.1 pozvolqgt dat\ bolee
syl\nug formu utverΩdenyq 2.1.
UtverΩdenye 2.1*. Pust\ X ∈ � y 1 ≤ card X < ∞. Tohda X yzometryç-
no vloΩymo v R , esly y tol\ko esly dlq X suwestvuet edynstvennaq dya-
metral\naq para.
Sledugwee utverΩdenye pokaz¥vaet, çto dlq koneçn¥x X ∈ � yzometry-
çeskaq vloΩymost\ v R πkvyvalentna suwestvovanyg toçky p ∈ X, dyamet-
ral\noj dlq vsex A takyx, çto p ∈ A ⊆ X.
UtverΩdenye 2.2. Pust\ ( X, d ) — koneçnoe nepustoe metryçeskoe pros-
transtvo. Tohda vloΩymost\ v R πkvyvalentna tomu, çto X ymeet M -
svojstvo y suwestvuet toçka p ∈ X, qvlqgwaqsq dyametral\noj dlq lgboho
B � p, t. e. takaq, çto ravenstvo
diam B = max { d ( p, x ) : x ∈ B } (2.9)
v¥polneno dlq lgboho B ⊆ X, soderΩaweho toçku p.
Dokazatel\stvo. Esly X — podmnoΩestvo R, to v kaçestve p moΩno
vzqt\ naymen\ßee çyslo yz X, çto, oçevydno, harantyruet (2.9). Pust\ teper\ X
ymeet M-svojstvo y suwestvuet p ∈ X takoe, çto (2.9) v¥polneno, kak tol\ko
p ∈ B ⊆ X. Yspol\zuq M-svojstvo, ubeΩdaemsq, çto suwestvuet edynstvennaq
toçka b ∈ B, dlq kotoroj diam B = d ( p, b ). Teper\ yskomoe yzometryçeskoe
vloΩenye f : X → R stroytsq, kak v utverΩdenyy 2.1.
Opyßem vse vozmoΩn¥e vloΩenyq metryçeskyx prostranstv v R.
Lemma 2.2. Pust\ ( Y, d ) — metryçeskoe prostranstvo, yzometryçno vlo-
Ωymoe v R, card Y ≥ 2, a x1 y x2 — razlyçn¥e toçky yz Y. Tohda dlq lgbo-
ho yzometryçeskoho vloΩenyq f : Y → R y lgboho x ∈ Y systema
| t – t1 | = d ( x, x1 ),
(2.10)
| t – t2 | = d ( x, x2 ),
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
OTNOÍENYE LEÛAT| MEÛDU Y YZOMETRYÇESKYE VLOÛENYQ … 1325
hde ti : = f ( xi ), i = 1, 2, ymeet edynstvennoe reßenye t = f ( x ). Obratno, esly
t1
, t2 ∈ R y | t1 – t2 | = d ( x1
, x2 ), to dlq lgboho x ∈ X systema (2.10) ymeet
edynstvennoe reßenye t ∈ R, a otobraΩenye Y � x � t ∈ R est\ yzometry-
çeskoe vloΩenye.
Dokazatel\stvo lemm¥ osnovano na tom, çto lgb¥e dve razlyçn¥e „sfer¥” v
R yly ne peresekagtsq, yly yx pereseçenye soderΩyt rovno odnu toçku.
Teper\ ustanovym neobxodym¥e y dostatoçn¥e uslovyq vloΩymosty v R
proyzvol\noho X ∈ �.
UtverΩdenye 2.3. Dlq lgboho metryçeskoho prostranstva ( X, d ) sledu -
gwye predloΩenyq qvlqgtsq πkvyvalentn¥my:
i) ( X, d ) yzometryçno nekotoromu podmnoΩestvu prqmoj R;
ii) ( X, d ) ymeet M -svojstvo y dlq lgb¥x t, s > 0 ne soderΩyt nykakoho
çet¥rextoçeçnoho podmnoΩestva, yzometryçnoho A t s4( ), ;
iii) esly A ⊆ X y card A ≤ 4, to A yzometryçno vklad¥vaetsq v R.
Dokazatel\stvo. Budem provodyt\ dokazatel\stvo po sxeme
i) � iii) � ii).
Ymplykacyq i) ⇒ iii) oçevydna. RassuΩdenyq, pryvedenn¥e v prymere 1.1,
pokaz¥vagt çto y ymplykacyq iii) ⇒ ii) qvlqetsq ystynnoj. PredpoloΩym te-
per\, çto predloΩenye iii) ystynno, y dokaΩem predloΩenye i). Budem sçytat\,
çto card X ≥ 4, tak kak v protyvnom sluçae ystynnost\ predloΩenyq i) oçevyd-
na. Zafyksyruem paru razlyçn¥x toçek x1
, x2 ∈ X y paru çysel t1
, t2 ∈ R, dlq
kotor¥x | t1 – t2 | = d ( x1
, x2 ). V sylu predloΩenyq iii) trojka x, x1
, x2 yzo-
metryçno vklad¥vaetsq v R . Sledovatel\no, v sootvetstvyy so vtoroj çast\g
lemm¥ 2.2 suwestvuet edynstvennoe t = t ( x ) ∈ R, dlq kotoroho
| t – t1 | = d ( x, x1 ) y | t – t2 | = d ( x, x2 ). (2.11)
Proverym, çto otobraΩenye X � x � t ( x ) ∈ R qvlqetsq yzometryçeskym
vloΩenyem, t. e.
| t ( x3 ) – t ( x4 ) | = d ( x3
, x4 ) (2.12)
dlq vsex x3
, x4 ∈ X. Poslednee ravenstvo sleduet yz (2.11), esly par¥ x3 , x4 y
x1
, x2 ymegt xotq b¥ odnu obwug toçku. Esly vse çet¥re toçky x1 , x2 , x3 , x4
razlyçn¥, to dlq dokazatel\stva (2.12) dostatoçno poloΩyt\ Y = { x1 , x2 , x3 ,
x4 } vo vtoroj çasty lemm¥ 2.2.
Ostalos\ proveryt\ ymplykacyg ii) ⇒ iii). PredpoloΩym, çto predloΩenye
ii) ystynno, a predloΩenye iii) loΩno. Tohda najdetsq A ⊆ X s card A ≤ 4, ne
vloΩymoe v R. V sylu M-svojstva lgboe A s card A ≤ 3 vloΩymo v R. Sle-
dovatel\no, suwestvuet ne vloΩymoe v R çet¥rextoçeçnoe A = { x1 , x2 , x3 , x4 }.
V sylu sledstvyq 2.3 moΩno sçytat\, çto
diam A = d ( x1 , x2 ) = d ( x3 , x4 ). (2.13)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1326 A. A. DOVHOÍEJ, D. V. DORDOVSKYJ
Pust\ s : = d ( x2 , x3 ), t : = d ( x3 , x1 ), s1 : = d ( x4 , x1 ), t1 : = d ( x4 , x2 ), r : = d ( x1 ,
x2 ) = d ( x4 , x3 ) (sm. rys. 1), polahaem r1 = r.
Yz M-svojstva y (2.13) poluçaem ravenstva
r = t + s = t1 + s1 = t + s1
,
otkuda t = t1 y s = s1
. Sledovatel\no, { x1 , x2 , x3 , x4 } yzometryçno A t s4( ), ,
çto protyvoreçyt predpoloΩenyg ob ystynnosty predloΩenyq ii).
UtverΩdenye dokazano.
Analyzyruq proverku ystynnosty ymplykacyy iii) ⇒ i) v dokazatel\stve ut-
verΩdenyq 2.3, poluçaem sledugwyj pryznak vloΩymosty metryçeskyx pros-
transtv v R.
Lemma 2.3. Pust\ ( X, d ) — metryçeskoe prostranstvo, card X ≥ 4 y x1 ,
x2 — fyksyrovann¥e toçky yz X. Tohda esly lgboe çet¥rextoçeçnoe podmno-
Ωestvo A ⊆ X, soderΩawee x1 y x2 , yzometryçno vklad¥vaetsq v R, to y
samo X vklad¥vaetsq v R.
Esly A — ohranyçennoe podmnoΩestvo metryçeskoho prostranstva ( X, d ),
ymegweho M-svojstvo, a { x1 , x2 }, { y1 , y2 } — dyametral\n¥e par¥ dlq A, to
lybo { x1 , x2 } = { y1 , y2 }, lybo { x1 , x2 } ∩ { y1 , y2 } = ∅, tak kak v protyvnom
sluçae diam ( { x1 , x2 } ∪ { y1 , y2 } ) > diam A.
∏to nablgdenye pryvodyt k sledugwej lemme.
Lemma 2.4. Pust\ ( X, d ) — metryçeskoe prostranstvo, ymegwee M-
svojstvo. Esly card X = 5, to lybo najdetsq rovno odna dyametral\naq para
{ x1 , x2 } ⊆ X, lybo takyx par rovno dve y ony ne peresekagtsq.
Dejstvytel\no, esly dan¥ try proyzvol\n¥x dvuxtoçeçn¥x podmnoΩestva
mnoΩestva X, to po krajnej mere dva yz nyx peresekagtsq.
Lemma 2.5. Pust\ X = { a, b, c, p, f } — pqtytoçeçnoe metryçeskoe pros-
transtvo s metrykoj d, ymegwee M -svojstvo. Tohda X yzometryçno
vklad¥vaetsq v R.
Dokazatel\stvo. Budem sçytat\, çto
diam X = d ( a, f ).
Pust\ d ( a, f ) > d ( x, y ) dlq vsex dvuxtoçeçn¥x mnoΩestv { x , y } ⊆ X, otlyçn¥x
ot a, f, y B — çet¥rextoçeçnoe podmnoΩestvo X takoe, çto { a, f } ⊆ B. Tohda
dlq lgb¥x t, s > 0 B ne soderΩyt A t s4( ), . Sledovatel\no, po utverΩdenyg
2.3 B vloΩymo v R, a prymenqq lemmu 2.3, vydym, çto y samo X vloΩymo v R .
Znaçyt, esly X ne vloΩymo v R , to suwestvuet ewe odna dyametral\naq para
{ x2 , x2 } ⊆ X, pryçem po lemme 2.4 { x2 , x2 } ∩ { a, f } = ∅, y takaq para edyn-
stvenna. Budem sçytat\, çto { x2 , x2 } = { b, p }. Çet¥rextoçeçnoe prostranstvo
{ a, b, p, f } ne vloΩymo v R, a znaçyt, po utverΩdenyg 2.3, ono yzometryçno
A t s4( ), dlq nekotor¥x t, s > 0.
Takym obrazom poluçaem systemu ravenstv
d ( a, b ) = s, d ( b, f ) = t, d ( a, c ) = u, d ( b, c ) = x,
d ( a, p ) = t, d ( p, f ) = s, d ( f, c ) = v, d ( c, p ) = y, (2.14)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
OTNOÍENYE LEÛAT| MEÛDU Y YZOMETRYÇESKYE VLOÛENYQ … 1327
d ( a, f ) = s + t, d ( b, p ) = s + t, d ( a, f ) = u + v, d ( b, p ) = x + y,
Rys. 2
pryçem
x + y = u + v = s + t = diam X (2.15)
(sm. rys. 2, na kotorom pryveden¥ podprostranstva { a, b, p, f }, { a, c, f } y { b, c,
p }). Pry sdelann¥x v¥ße predpoloΩenyqx podprostranstva { a, p, c, f } y { a,
b, c, f } vloΩym¥ v R v sylu predloΩenyj i) y ii) yz utverΩdenyq 2.3. Rassmot-
rym vloΩenyq { a, p, c, f } → R. VozmoΩn¥ sledugwye varyant¥ (rys. 3):
a1
) c leΩyt meΩdu a y p,
a2
) c leΩyt meΩdu p y f.
Analohyçno dlq vloΩenyq { a, b, c, f } → R (rys. 4) ymeet mesto odno yz dvux:
b1
) c leΩyt meΩdu a y b,
b2
) c leΩyt meΩdu b y f.
Esly realyzovano soçetanye a1
) y b1
), to yz a1
) y (2.14) poluçaem y + u = t y y +
+ x = s + t, znaçyt, t – u = s + t – x y, sledovatel\no, x = u + s. Krome toho, yz
b1
) ymeem u + x = s, t. e. x = 2u + x, çto nevozmoΩno. Pust\ realyzovano soçe-
tanye a1
) y b2
). Tohda y + u = t = x + v, t. e. y = x + v – u, a yz (2.15) y = u + v –
x. Sledovatel\no, x + v – u = v – x + u, t. e. u = x, no, kak otmeçeno v¥ße, (2.14)
y a1
) vlekut ravenstvo x = s + u, çto protyvoreçyt sootnoßenyqm s = d ( a, b ) >
0.
Oçevydnaq modyfykacyq provedenn¥x rassuΩdenyj pokaz¥vaet, çto soçeta-
nyq a2
) y b1
), a2
) y b2
) toΩe vedut k protyvoreçyqm.
Rys. 3
Rys. 4
Dokazatel\stvo teorem¥ 1.1. Pust\ X ∈ �. VozmoΩn¥ try sluçaq:
card X ≥ 5, card X ≤ 3 y card X = 4. V pervom sluçae po lemme 2.5 X yzometryçno
vloΩymo v R. Pry card X ≤ 3 vloΩymost\ X tryvyal\no sleduet v sylu M-
svojstva.
Rassmotrym sluçaj card X = 4. Pust\ X = { x1 , x2 , x3 , x4 } y diam X = d ( x1 ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1328 A. A. DOVHOÍEJ, D. V. DORDOVSKYJ
x2 ). Esly d ( x1 , x2 ) > d ( x3 , x4 ), to v sylu sledstvyq 2.1 X yzometryçno vloΩy-
mo v R. Yz ravenstva d ( x1 , x2 ) = d ( x3 , x4 ) sleduet, çto X yzometryçno
A t s4( ), (sm. dokazatel\stvo utverΩdenyq 2.3).
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ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
|
| id | umjimathkievua-article-3103 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:36:17Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e5/f4da11c8428461caa10f30513d8ff0e5.pdf |
| spelling | umjimathkievua-article-31032020-03-18T19:45:28Z Betweenness relation and isometric imbeddings of metric spaces Отношение лежать между и изометрические вложения метрических пространств Dovgoshei, A. A. Dordovskii, D. V. Довгошей, А. А. Дордовский, Д. В. Довгошей, А. А. Дордовский, Д. В. We give an elementary proof of the classical Menger result according to which any metric space X that consists of more than four points is isometrically imbedded into \( \mathbb{R} \) if every three-point subspace of X is isometrically imbedded into \( \mathbb{R} \). A series of corollaries of this theorem is obtained. We establish new criteria for finite metric spaces to be isometrically imbedded into \( \mathbb{R} \). Наведено елементарне доведення класичного результату X. Менгера про те, що будь-який метричний простір X, що складається більш ніж з чотирьох точок, ізометрично вкладається в \( \mathbb{R} \), якщо кожний триточковий підпростір X ізометрично вкладається в \( \mathbb{R} \). Отримано ряд наслідків з цієї теореми. Встановлено нові критерії ізометричної вкладеності в \( \mathbb{R} \) скінченних метричних просторів. Institute of Mathematics, NAS of Ukraine 2009-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3103 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 10 (2009); 1319-1328 Український математичний журнал; Том 61 № 10 (2009); 1319-1328 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3103/2954 https://umj.imath.kiev.ua/index.php/umj/article/view/3103/2955 Copyright (c) 2009 Dovgoshei A. A.; Dordovskii D. V. |
| spellingShingle | Dovgoshei, A. A. Dordovskii, D. V. Довгошей, А. А. Дордовский, Д. В. Довгошей, А. А. Дордовский, Д. В. Betweenness relation and isometric imbeddings of metric spaces |
| title | Betweenness relation and isometric imbeddings of metric spaces |
| title_alt | Отношение лежать между и изометрические вложения метрических пространств |
| title_full | Betweenness relation and isometric imbeddings of metric spaces |
| title_fullStr | Betweenness relation and isometric imbeddings of metric spaces |
| title_full_unstemmed | Betweenness relation and isometric imbeddings of metric spaces |
| title_short | Betweenness relation and isometric imbeddings of metric spaces |
| title_sort | betweenness relation and isometric imbeddings of metric spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3103 |
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