Page 1
Equilateraldimensionoftherectilinearspace
JackKoolen?MoniqueLaurent?andAlexanderSchrijver
CWI?Kruislaan????????SJAmsterdam?TheNetherlands
DedicatedtoJ?J?Seidelontheoccasionofhis??thbirthday?
Abstract
Itisconjecturedthatthereexistatmost?kequidistantpointsinthek?
dimensionalrectilinearspace?Thisconjecture hasbeenveri?edfork???we
showhereitsvalidity indimensionk???Wealsodiscussanumberof related
questions?Forinstance?whatisthemaximumnumberofequidistantpoints
lyinginthehyperplane?
P
k
i??
x
i
???Ifthisnumberwouldbeequaltok?
thentheaboveconjecturew ouldfollo w?Wesho w? how ever?that thisn umber
is?k??fork? ??
?Introduction
???Theequilateralproblem
F ollowingBlumenthal?B?? ??a subsetX ofametric spaceM is said tobee quilateral
?ore quidistant? ifanytwodistinctp oin tsofX areatthe samedistance?then?the
e quilateral dimensione?M?ofMisde?nedasthemaximum cardinalityofan
equilateralsetinM?
Equilateralsetshavebeenextensivelyinv estigatedintheliteratureforanum?
berofmetricspaces?includingspherical?
e??
h
??
yp erb
?
olic?ellipticspacesandrealnormed
spaces?Theirstructureisw ellunderstoodintheEuclidean?sphericalandhyper?
bolicspaces?cf??B????andresultsaboutequiangularsetsoflines aregivenby
van LintandSeidel?vLS???andLemmensandSeidel?LS????Aswewillseebelow
someboundsareknownfortheequilateraldimensionofanormedspacebutits
exactvalueisnotknown?exceptfor theEuclidean and?
?
?norms??In thispaper
wefocusontherectilinearspace?
?
?k??thatis?therealspaceR
k
equippedwith
the?
?
?norm??Forx?R
k
?its?
?
?norm iskxk
?
?
P
k
i??
jx
i
j??Clearly?
?
?k?k
astheunitvectorsand their opposites formanequilateral set?Itisgenerally
b eliev ed?see?in particular? Kusner?GK????that?kistherightv alueforthe
equilateral dimension?
Conjecture ??F ore achk???e??
?
?k ????k?
Thisconjecture hasbeenshowntoholdfork?? ?BCL?? ??Our mainresult in
thispaperisto showitsv alidityinthe nextcasek? ???Cf?Theorem ???
?
Page 2
What playsanessen tial role inourpro ofis thefactthat theequilateralproblem
intherectilinear space?
?
?k?canbereformulated asadiscrete ???problem?which
permitsadirectsearch attacktotheproblem?namely?provingConjecture? for
giv enk reducestochec kingthenonexistenceofacertainsetsystemon?k??ele?
ments?Moreover?weform ulateastrongerversionofConjecture? ?cf?Conjecture
?? whichallo wsafurthersimpli?cationinthepro ofsinceitsu?cesto consider
certainsetsytems on?k??elements?insteadof?k????Thisreformulation is
presentedinSection?and thepro ofof Conjecture? inthecasek??isgiven in
Section ??
InSection ??wediscuss severalfurtherquestionsrelatedtotheequilateral
problemin therectilinearspace? Inparticular? whatisthe maxim umcardinality
of anequilateral setlyinginah yp erplane
P
k
i??
x
i
?? ofthe rectilinear spaceR
k
? ?Isitk?? What isthe maxim umn umb er ofpairwise touc hingtranslates ofak?
dimensional simplex? ?Isitk?????Calltwoconv exbodiestouching iftheymeet
buthavedisjointin teriors?? Do es everydesign onnpoin ts con tainanan tic hain
of sizen? Thesequestions are in somesense equivalentandap ositiveanswer
toany ofthemwouldimplyapro of of ourbasicConjecture??cf?Proposition
???? Howev er?exceptforsmallk orn?theanswersproposedabove arenotcorrect?
Indeed?foranyn? ??thereexistsadesign onnpoin ts having noan tic hainof size
n?for anyk???thereexistk??pairwisetouc hingtranslates ofak?dimensional
simplex ?cf? Proposition????
???Relatedgeometricquestions
Theproblemofdeterminingthe equilateraldimension ofanormedspaceV arises
inparticular when studyingsingularities ofminimalsurfaces andnetw orks ?cf?
?M???FLM??? LM?????This problem hasthe follo winginterestinggeometric in?
terpretation?LetK denotethe unit ballofthe normed spaceVand lett?K?
denotethethe maxim umn umb er of translates ofK thatpairwise touch? called
thetouchingnumberofK?Givenx
?
?????x
n
?V? the setfx
?
?????x
n
gisequilateral
withcommondistance? if andonlyif thetranslatedbodiesK?x
?
?????K?x
n
arepairwisetouc hing?Hence? theequilateraldimensione?V?ofthenormed space
V isequal to thetouching n umbert?K?ofitsunitballK?
Upp erb ound?A simplevolumeargument showsthate?V???
k
ifVisk?
dimensional?indeed? A??
S
n
i??
?K?x
i
?iscontainedintheballofcenterx
?
and
radius??Asnotedin?FLM????thisupperboundcanbere?nedto?
k
byobserving
thatAhasdiameter?andusingthe isodiametricinequalitywhichstatesthatthe
volume ofabodywithdiameter??islessthanorequaltothevolumeoftheunit
ball?The?
k
upperboundhadbeenobtainedearlierbyPett y?P???whoshow ed
the follo wingstructuralc haracterizationforequilateralsets?AsetX?R
k
is
equilateralwithrespecttosomenormifandonlyifXis anantipo dalset ?thatis?
forany distinctpointsx?x
?
?Xthereexisttwoparallelsupp ortinghyperplanes
?
Page 3
H?H
?
forXsuchthatx?H?x
?
?H
?
??the?
k
bound nowfollo wsfrom the
factestablished in?DG??? that anantipo dalset inR
k
has atmost?
k
?
poin ts?
Clearly? the?
k
upp erbound isattainedfor the?
?
?norm?asf???g
k
isequilateral??
moreov er? anequilateralsetofsize?
k
existsonly whentheunitballK isa?nely
equivalent to thek?cube?P????
Low erbound?Petty?P??? showsthatone can?ndfourequidistantpoin tsin
anynormed spaceofdimension? ??It isstill anopenquestion todecidewhether
one can ?nd anequilateralset ofcardinalityk?? inanormedspace ofdimension
k???cf? ?M??? LM???S??? or?T??? ?problem ????? page??????Note? how ev er?
that theansw er isobviouslyp ositive forthe?
p
?norm ?ase
?
?????e
k
??a????a? form
an equilateralset? wheree
?
?????e
k
are the unitvectors andasatis?esja??j
p
the
?
?k???jaj
p
????In theEuclideancase?p? ???k?? isthe rightv alueforthe
equilateraldimension?B?? ??
Hadwiger?sproblem? Theequilateralproblem has interestingconnections to
sev eralotherproblems in combinatorialgeometry? Inparticular? it isrelated to
aclassicproblemp osedbyHadwiger?H???whichasksforthe maximumnumb er
m?K? oftranslatesofaconv exbo dyKthat allmeetK andhavepairwisedisjoin t
interiors? ?Seep? ??? in?DGK???forhistory? resultsandprecisereferences on
Hadwiger?sproblem?? It canbe sho wn thatm?K??H?K?? ??whereH?K? is
maxim umn umb eroftranslates ofK thatall touchK and havepairwisedisjoint
interiors?H?K? is known astheHadwigernumb er ?or translative kissingnumb er?
ofK? Inotherwords?whenK is centrallysymmetricwithassociatednormk?k?
H?K? is the maxim umn umbern ofvectorsx
?
?????x
n
satisfying?kx
i
k??and
kx
i
?x
j
k??foralli??j???????n? Thetouc hing andHadwigern umb ersare
relatedbytheinequalit y?t?K??H?K?? ??
LetKbeak?dimensionalconvexbody?thefollowingisknown?H?K???
k
??
?Hadwiger?H?? ??simplevolumecomputation??H?K???
k
??if andonlyifKis
aparallelotope?Gr ? unbaum ?G??b? fork?? and Groemer?G??a? forgeneralk ??
H?K???whenKisa??dimensionalconvexbodydi?erent fromaparallelogram
?G??a ??H?K??k
?
?k?Swinnerton?Dy er?SD????? Theprevious low erboundwas
recently improv edbyTalata?T??? who show edtheexistence ofaconstantc??
suchthatH?K???
ck
for anyk ?dimensionalconvexbo dyK?Determiningthe
Hadwigernumberofthek?dimensionalEuclideanballB
k
isa longstandingfamous
op enproblemwhichhassurgedintensiveresearch?inparticular?itisknownthat
H?B
k
??k
?
?k fork???TheHadwigernumberof thetetrahedronwasrecently
shown tobe equalto???Talata?T?????
Otherrelatedcombinatorialproblemsareinvestigatedin?FLM???S???S????
Forinstance?ifx
?
?????x
n
?R
k
areunitvectors?withrespecttosomenorm?
satisfyingkx
i
?x
j
k??foralli??j?thenn??
k ??
? moreover?n??kif?belongs
totherelativ e in terioroftheconvexhullof thex
i
?s?orifk
P
i?I
x
i
k?? forallI?
???n??Furthergeometricquestions?like theproblemof?ndinglargeantichainsin
?
Page 4
designs or theproblem ofdeterminingthe maximumnumb er ofpairwisetouc hing
translates ofa simplex? willbediscussed inSection ??
? Reform ulating theequilateral problem intherecti?
linear space
Wepresent heresomereform ulationsof theequilateralproblem in therectilinear
space?
?
?k? intermsof setsystems?
We introduce some de?nitions?Giv enX?fx
?
?????x
n
g?R
?
? leta
?
?????
a
p
denotethedistinctv alues takenbyx
?
?????x
n
and set
S
q
??fi????n?jx
i
?a
q
g forq??????? p?
Then?B?X?denotes thew eighted set system onV ?? ???n?consisting of the setsS
q
withw eight?
S
q
??a
q
?a
q??
forq???????p ?settinga
?
????? Then?S
p
?????S
?
and thefollowing holdsfori??j?V?
??? ?i?x
i
?
X
S?B?X?ji?S
?
S
? ?ii?jx
i
?x
j
j?
X
S ?B?X ??jS ?fi?j gj??
?
S
?
Generally? givenX?fx
?
?????x
n
g?R
k
?
?we letB?X? denote theweigh ted set
system de?nedasthe union ofthekw eighted setsystemsB?fx
?
?h??????x
n
?h?g?
forh???????k?Then?B?X? canbe cov eredbykchains andthe follo wingholds
fori??j?V?
??? ?i?e
T
x
i
?
X
S ?B?X?ji?S
?
S
??ii?kx
i
?x
j
k
?
?
X
S ?B?X ??jS ?fi?jgj??
?
S
?
When allv ectors inX arenonnegative inte gral?B?X? canbeview edasam ultiset
ifwereplaceaw eighted setSwithw eighta?ap ositiveinteger?byao ccurences
ofS? Note thatthecorrespondenceX ??B?X?ismany?to?one ?asthere maybe
several wa ysofpartitioningasetsystem in tochains??Forinstance? consider
M
?
?
?
?
???
???
???
?
A
?M
?
?
?
?
???
???
???
?
A
?A?
?
?
??????
??????
??????
?
A
and letX
?
?X
?
denotethe setsinR
?
whosepointsaretherowsofM
?
andM
?
?
respectively?Then?B?X
?
??B?X
?
? isthem ultiset givenby thecolumns ofA?
X
?
andX
?
correspondtotwodistinct partitionsof thecolumns ofAin tochains?
namelywith partsf???g?f???g?f???g?andwithpartsf???g?f?g?f?????g?
Giv enasubsetS?V? the cut??S?isthevector off???g
?
n
?
?
de?nedby??S?
ij
?
? ifandonlyifjS?fi?j gj??for??i?j?n?Let??
n
denote theall?
onesvectorinR
?
n
?
?
?AcutfamilyS issaid tobe nestedifitsmemberscanbe
ordered as??S
?
????S
?
????????S
m
?in suchaway thatS
?
?
?S
?
?
?????S
?
m
where
?
Page 5
S
?
j
?fS
j
?VnS
j
g for eachj???????m?S is saidtobek ?nested
?
if itcanbe
decomposed asa union ofknestedsubfamilies?A cutfamilyS issaid tobe
equilater al if thereexistpositivescalars?
S
???S??S?for which thefollowing
relation holds?
?????
n
?
X
??S??S
?
S
??S??
Clearly? ??? holdsifand onlyif the rows of thematrix whosecolumns arethe
v ectors?
S
?
S
atten
???S??S? form anequilateral set? ?Giv enasetS?V??
S
?f???g
V
denotes itsc haracteristicv ector de?nedby?
S
i
?? if andonly ifi?S? fori?V ??
F orinstance?
n
X
i??
??i?? ???
n
?
which sho wsthatthe cutfamilyf??i?ji???????ng isequilateral? this cutfamily
is called thetrivial cutfamily?Finally? notethat in???we canassume that the
scalars?
S
arerationaln umbers? similarly? when looking for equilateralsetswe can
restrictourtion tononnegative integral ones?To summarize?we havesho wn?
Proposition ??The followingassertions aree quivalent?
?i? There exists ane quilateral set in?
?
?k? ofc ardinalityn?
?ii? There existsa multisetB on???n?which isc overed bykchainsandsatis?es
jfS?B?jS?fi?j gj??gj?r foralli??j?V? forsomer???
?iii? Thereexistsak ?nestede quilateral cutfamily onnelements?
For smallnone canmake anexhaustive search of all the equilateral cutfamilies
onnpoints?F orinstance? thetrivial cutfamily isthe onlyequilateralcutfamily
on?poin ts andforn?? thefollowing result canbe easilyveri?ed?
Lemma ??F orn???any dec omposition??? has theform?
??
?
??
?
X
i??
??i???
?
?
???
?
X
i??
? ??i?where????
?
?
?
?
As isw ell known?the minim umn umb er ofc hainsneededforco veringaset system?moregen?
erally?a partiallyordered set?isequal to themaxim umcardinality of an antichain ?byDilworth?s
theorem?and canbedetermined inpolynomialtime?usinga maximum?owalgorithm??Fleiner
?F???has giv ena minimaxformula for theminimumnumberk ofnested subfamiliesneeded to
cover a cutfamily ?moregenerally? forasymmetricp oset? andsho wn that itcanbe determined
inpolynomialtime?byareduction to thematchingproblem??
?
Page 6
ofConjecture??
Lemma ??Consider theassertions?
?i?A nyk ?nestedequilater al cutfamilyon?k??elements istrivial?
?ii?A nyk?nestedequilater alcut family on?kelements istrivial?
?iii? There do esnotexistak?nestede quilater al cutfamily on?k??elements?
i?e??e??
?
?k ????k?
Then??i??? ?ii??? ?iii??
Pr oof??i??? ?ii? LetSbeak ?nestedequilateral cutfamily onV?jVj??k?
AssumethatS isnot trivial andlet??S??S with??jSj??k? ??F or eachi?V?
theinducedcut familyonVnfig istrivialwhich impliesthatS??Vnfig?j??
or?k? ?? Choosingi?VnS?we obtain thatjSj??k?? andc ho osingi?S
thatjSj? ??Therefore?k? ?? Inview ofLemma ??Scontains the threecuts
?????????????? ???? con tradicting theassumptionthatS is ??nested? Thepro offor
implication?ii??? ?iii?is similarand th usomitted?
Giv enx
?
?R
k
and ?? ?? thesetX ??fx
?
? ?e
i
ji???????kg is ob viously
equilateral? any set ofthis form is calledatrivialequilateralsetinR
k
? Given
x?y?z?R
k
?their median isthep ointm?R
k
whosehth co ordinateis themedian
v alue ofx
h
?y
h
?z
h
forh???????k? Asisw ellkno wn? the medianm is theunique
point lying onthethree geo desicsb ew eenanytwo ofthep oints x?y?z? thegeo desics
beingtak enwithrespect tothe?
?
?distance?and the geodesicb etweenx andy
consistingof allpoin tsu?R
k
satisfyingkx?yk
?
?kx?uk
?
?ku?yk
?
?Weno w
reform ulateLemma? ?ii? inmore geometricterms?
Lemma ??Consider theassertions?
?i? A nyk ?nestedequilater al cutfamily on?kelements istrivial?
?ii?Anyequilater alset inR
k
ofc ar dinality?k is trivial?
?iii?IfX isanequilater al setinR
k
ofc ardinality?k?andwithcommon distance
?? then there existsx
?
?R
k
such thatkx
?
?xk
?
?? forallx?X?
Then??i????ii????iii??
Proof? ?i????ii?LetX?R
k
be equilateralofcardinality?k andwithcommon
distance ??up totranslationwecan supposethat min?x
i
?h?ji????????k???
forh???????k? LetB?X?denote the associatedweigh tedsetsystemasexplained
earlierinthissection?By?i??weknowthat everysetinB?X? isa singleton orthe
complementofasingleton?Therefore?we ?nd?uptop ermutationon???????k?
?
Page 7
thatB?X?consists of the setsf?i??g andVnf?ig fori???????k?each with
multiplicity ?? Usingrelation????i?? thisimpliesthatXconsists ofthep oin tse?e
i
?i???????k ??wheree is theall?onesvector?that is?X istrivial?
?ii??? ?iii?holdstrivially?
?iii??? ?i? LetSbeak ?nestedequilateral cut familyon?kp oin ts? Then?a
suitablechoice ofS orVnS foreach cut??S??S yieldsaw eightedset systemB
onV? ????k?which is cov eredbykc hains andsuchthat??
n
?
P
S ?B
?
S
??S ?? Let
X?fx
?
?????x
?k
g?R
k
denoteanequilateral setcorresp ondingtoB ?de?nedusing
????i? anda givenpartition ofB intokchains?? By ?iii??we obtainthat any three
distinctp oin tsofX have thesamemedian? Therefore?foreveryh???????k? the
vector?x
?
?h??????x
?k
?h??isof theform a?
i
? b?
Vnj
wherei??j?V and a?b? ??
F rom thiswe see thatS is thetrivialcutfamily?
Tosummarize?we can formulate thefollowingconjectures?
Conjecture ??A nyk ?nestede quilateral cutfamilyon?k?? elements istrivial?
Conjecture??A nyk ?nestede quilateral cutfamily on?kelementsistrivial?
Equivalently?anyequilater alsetinR
k
ofc ardinality?k is trivial?
Proposition??Conje cture???Conje cture??? Conjecture??
Conjecture?holds fork???trivial? andfor k????BCL?????We show that
italso holds fork? ??thepro ofisdelay edtill Section??
Theorem??Conjecture?holdsfork???
?Connections toothergeometricproblems
???T ouc hingcross?p olytopes
Let?
k
?fx?R
k
?kxk
?
??gdenote the unitball ofthek?dimensionalrectilinear
space? ?
k
isalso known as thek?dimensional cross?polytop e?As men tionedinthe
in troduction?theequilateral dimension of?
?
?k?isequal tothe touc hingn umb er
of?
k
?A morerestrictivequestion is todetermine the maximumn umb erofpair?
wisetouching translatesof?
k
thatshareacommonp oin t?Thisquestion canbe
answered easily?
?
Page 8
Lemma ??? Themaximum numb erofp airwisetouching translates ofthecr oss?
p olytope?
k
sharingac ommonpoint ise qualto?k?
Proof?Clearly??k isa low erbound ?since the?
k
?e
i
?s?i???????k? allmeet at
theorigin?? The fact that?k isan upp erb ound follows from results in?HH??? F???
on the?
?
?emb eddingdimension oftrees? ?Itcanalsobec heck ed directly usingthe
samereasoning as for theimplication ?iii??? ?i?ofLemma ???
Hence?we?nd again thatConjecture? holdsif one canshow thatthere areat
mostn??k pairwisetouc hingtranslates of?
k
having no commonp oint?that is?
ifConjecture?holds??this is?infact?the pro oftec hnique usedin ?BCL??? inthe
casek? ???
Let usobservethat touchingtranslates ofthe cross?polytope enjoyastrong
Hellyt ype prop erty? Namely? ifB
i
???
k
?x
i
?i???????n? aren pairwise
touc hingtranslatesof?
k
?thenB
i
?B
j
?B
h
isreduced toa singlep oint ?the
median ofx
i
?x
j
?x
h
? for anydistinct i? j?h? ???n??therefore?
T
n
i??
B
i
??? if and
only ifB
?
?B
?
?B
?
?B
i
??? for alli???????n?
???An tic hainsin designs andtouc hingsimplices
We present heresomevariationson theequilateralproblem inthe rectilinearspace?
dealing withequilateralsets onah yperplane? antichains indesignsandtouc hing
simplices?
A ?rstv ariationasksfor the maximumcardinalityh?k? of an equilateralset
X?R
k
lying inah yperplaneH
r
??fx?R
k
je
t
x?rg ?forsomer?R??
?Recallthate denotesthe all?onesvector?? Clearly?h?k??k?considering thek
unitvectors??
Thew eighted setsystems B?X?corresponding to integralequilateral setsX
lying inah yperplaneH
r
leadnaturally to thenotion of designs?Recallthat?giv en
p ositive in tegersr???am ultisetBonV? ???n? iscalled an?r????designif ev ery
p ointi?Vb elongstor memb ers ?bloc ks?ofB andanytwodistinctp oin tsi?j?V
belong to? commonmembers ofB? AnantichaininB isasubset ofBwhose
memb ers arepairwiseincomparable? Leta?n?denote themaxim umintegersuch
thateverydesignonnp oin ts hasan antichainofcardinalitya?n??equivalently?by
Dilworth?stheorem?a?n? istheminimumtakenov eralldesignsBonnpoints of
theminim umn umb erofchains needed to cov erB ?? Clearly?a?n??n ?considering
thedesign consisting ofall singletons??Equalitya?n??nw ould meanthatev ery
designonnp ointshasanantichainofsizen?Itiswell?known thateverydesign
onnpointscontainsatleastndistinct bloc ks?cf??dBE????thisfactisalsoknown
asFisher?sinequality??Therefore?anypairwisebalancedinc omplete design?that
is?a designBwhoseblocks allhavethesamecardinality?containsobviouslyan
antichainofsizen?
?
Page 9
Calla designB ona setV self?complementary if? for everyB?V? theset
B andits complementVnB app earwith thesamemultiplicity inB? Denoteby
a
?
?n? themaxim umcardinality ofan an tichain inaself?complemen tarydesign on
np oin ts?Hence?a?n??a
?
?n?? n?
Finally?weconsider thetouc hingn umb ert??
k
? of thek?dimensional regular
simplex?
k
?thatis? themaxim umn umb er ofpairwise touching translatesof?
k
??
We hav e?t??
k
??k? ??
?
Indeed? induction onk sho ws easily theexistenceof
k?? translates of?
k
thatare pairwise touching andsharea commonp oin t? ?Cf?
Remark????
Proposition ??? The followingholdsfor integersk?n???
?i?h?k??n??a?n??k?
?ii?h?k??t??
k??
??
?iii?a?n???k??e???k ???n?
?iv?a
?
?n? ????k????e??
?
?k ???n?
Pr oof??i? LetX?fx
?
?????x
n
g?Z
k
?
andletB?X?be its associatedm ultiset on
V? ???n?? Usingrelation????we deduce thatB?X? isa?r????design ifand onlyif
X is contained in theh yp erplaneH
r
andX isequilateral withcommon distance
?? ??r???? Moreov er?B?X? is cov eredbykchainsbyconstruction?This sho ws
?i??
?ii?We need the following notation?Giv enx?y?R
k
? letx?y denotethevector
ofR
k
whoseh?thcomponent isequal to max?x
h
?y
h
??
?
?
?x
h
?y
h
?jx
h
?y
h
j? for
h???????k?We hav e?
e
T
?x?y??
?
?
?e
T
x?e
T
y?kx?yk
?
??
LetS
?
?????S
n
bepairwisetouchingtranslates of theregular?k? ???dimensional
simplex?We cansupposethat theS
i
?s arealltranslates ofthe simplexS
?
??
fx?R
k
jx???e
T
x??g andthat theylie intheh yperplaneH
?
? Then?
S
i
?S
?
?x
i
?fx?R
k
jx?x
i
?e
T
x??gwherethex
i
?slieinH
?
? AsS
i
?S
j
?
fxjx?x
i
?x
j
?e
T
x??g andS
i
?S
j
are touc hing?we deducethate
T
?x
i
?x
j
?? ??
whichimplies thatkx
i
?x
j
k
?
???Therefore?thesetfx
?
?????x
n
g isequilateral
inH
?
? Conv ersely? ifX?fx
?
?????x
n
g?R
k
isanequilateralsetwithcommon
distance? andlying inH
?
? thenthensimplicesS
i
??fx?R
k
jx?x
i
?e
T
x??g
?i???????n? are pairwisetouching?Thisshowsthath?k??t??
k??
??
We prove ?iii?and?iv?together?F orthis?letBbeam ultiseton???N?whichis
coveredbykchainsandsatis?es?
jfS?B?jS?fi?jgj??gj?r
?
Page 10
for alli??j? ???N ??We show that? ifa?n???kora
?
?n? ????k? ?? thenN?n
?recall Proposition??ii??? Say?B??
k
h??
B
h
whereeachB
h
isac hain?Without loss
ofgeneralitywe can suppose thatthe elementNb elongsto all setsS?B
?
?We
de?netwo newmultisetsB
?
on ???N? ??andB
??
on???N? in the follo wing manner?
B
?
??fS?BjN ??Sg?f???N?nSjN?Sg?
B
??
??B?f???N?nBjB?Bg?
Obviously?B
?
iscov eredby?? ??k?????k??c hainsandB
??
by?kc hains?
Moreov er? oncanverify thatB
?
isa?r?
r
?
??design onN??p oin ts andthatB
??
is
a?jBj? jBj?r??design onNp oin ts?Therefore?we ?ndN???n? ??i?e??N?n
whena?n???k? andN?n whena
?
?n? ????k? ??
Therefore?Conjecture?w ouldhold if one couldshow that ev erydesign onn
p oin ts hasan antic hainofsizen? One canshow that thelatter holdsforn? ??
how ever? for eachn? ?? one canconstructadesignB
n
onnp oints ha ving no
antichain of sizen ?cf?Proposition ??b elo w??F orn? ?? one can show thatB
?
is
the only designhaving noan tichainof size??unique uptoaddition of thefull set
??? ????Thisp ermitstoshowthatanydesign on?p oin ts has anan tichain ofsize
??Tosummarize? we hav e?
h?k??t??
k??
??k fork? ??a?n??n forn? ??
h?k??t??
k??
??k??fork? ??a?n??n?? forn? ??
a?n??k forh?k? ???n?h?k??
Inparticular?
a??????a??????h????t??
?
????
Moreov er?we havecheck edthat
a
?
?n??n forn???
Example ???We describe heretwodesignsB
n
onn????p ointswhichare
coveredbyn??chains? asw ellas theassociatedequilateralsetsinR
n??
?v ectors
aretherows of thearrays? ofcardinalityn?
B
?
?
?? ????
?????????? ????
????
????
????
????
????
??
Page 11
B
?
?
?????????? ?????????? ?????
?? ?? ??? ????????????
???????? ??????????
?????
?????
?????
?????
?????
?????
Another designon?p ointscov eredby?c hains?
??????????????????????????
???? ????
???? ???? ???? ????
?????
?????
?????
?????
?????
?????
Proposition ???F ore achn??? there existsadesign onnp ointswhich is
c overed byn??chains?
Pr oof? Usinginduction onn??weconstructa designB
n
onnp oin tswhich is
cov eredbyn??c hains and withparametersr
n
??
n
satisfying?
??? jB
n
j??r
n
??
n
andfig?B
n
for alli???????n???
DesignB
?
is as described inExample ??? itsatis?es ???? GivenB
n
satisfying ????
we letB
n??
consist of the following sets?B?fn??g forB?B
n
?f??????ng
rep eatedr
n
??
n
times and?fori???????n?fig repeated jB
n
j??r
n
??
n
times?
Then?B
n??
isa design with parametersr
n??
? jB
n
j??
n??
?r
n
? Moreov er?
jB
n??
j? jB
n
j??r
n
??
n
??n?jB
n
j??r
n
??
n
?
whichimpliesthat
jB
n??
j??r
n??
??
n??
??n????jB
n
j??r
n
??
n
????
Hence???? holdsforB
n??
? Finally?B
n??
canbe cov eredbync hains since onecan
assign thesingletonsfig?i???????n? ??tothen??c hainscov eringfB?fn??gj
B?B
n
g and putf??????ng andfng togetherinanewc hain?
Remark ??? Themaxim umnumb er ofpairwise touchingtranslates of the?k? ???
dimensional simplex thatshareacommon point isequal tok? ?Indeed?similarly
??
Page 12
to thepro ofofProposition ???one can showthat thereexistn touc hingtrans?
latesof?
k??
sharingacommonp oint if andonly ifthereexist anequilateral set
X?fx
?
?????x
n
g inah yperplaneH
r
ofR
k
such thatx
i
?x
j
isaconstantv ector
foralli??j?this inturn means thattheasso ciatedm ultisetB?X?consists of
copies ofV? ???n? and ofVni fori?V? which implies thatn?k sinceB?X? is
coveredbykc hains??
? Proof ofTheorem?
LetSbea cutfamily onV? Calltwocuts??S ????T?cr ossingif thefoursets
S?T?Vn S?VnT are pairwiseincomparable andcross?freeotherwise? in other
w ords?two cutsarecross?free ifand onlyif theyforma nestedpair? Givent?
jVj
?
?
a cut??S? is calledat?split ifS hascardinalityt orjVj?t? Giv ena subsetX?V?
letS
X
denote theinduced cut family onX?consisting of thecuts?S?X?XnS ??
In whatfollo ws?V?f???????g andS isassumedtobea nontrivial equilateral
cutfamily onV which is??nested? moreov er?wec ho osesuchSminimal with
resp ecttoinclusion?
IfX?V withjXj?? then?by Lemma ??S
X
either contains all??splits or
con tains no ??split? The?rststep of thepro of consistsofsho wingthat the former
alwa ysholds?
Proposition ???ForeveryX?VwithjXj???S
X
c ontainsall??splits?
Proof?Assume thattheresult from Proposition ??does notholdforsome subset
X?V? say?X ??f???????g? ByLemma??S
X
containsno ??split and? thus?S
X
contains allthethree ??splits onX? Hence?S canbepartitioned in to
S?S
?
?S
?
?S
?
?S
?
where allcuts inS
?
?resp?S
i
?i????? ??areofthe form??S? ?resp?? ??iS?? for
someS?W ??VnX?f?????g andS
i
??? fori????? ?? Notethatanytwo
cutsbelonging todistinctfamiliesS
i
?S
j
?i??j????? ??are crossing?Therefore?
asS is ??nested?we deduce that
???at leasttwo ofthefamiliesS
?
?S
?
?S
?
arenested?
AsS isequilateralwehav e?
??
?
?
X
S?W
?
?
S
??S??
X
i??????
X
S?W
?
i
S
? ??iS?
forsomenonnegativescalars?
?
S
??
i
S
?Sconsistingof thosecutsha vingapositive
??
Page 13
co e?cient?F orx??y?W andi??????? ??set
?
i
?x? ??
X
S?Wjx?S
?
i
S
??
i
?x? ??
X
S?Wjx??S
?
i
S
??
i
?xy? ??
X
S?Wjx?y?S
?
i
S
?
By evaluating coordinatewise the righthand side ofthe above decompositionof
??
?
we ?nd therelations?
????
?
?x???
i
?x???
i
?x??
?
?
fori?????? andx?W?
????
?
?xy??
X
i??????
?
i
?xy??
?
?
forx??y?W?
Weclaim thatifS
i
isnested forsomei????? ??then
S
i
?f? ??i??? ??iW?g?
Indeed?assume thatS
i
consists ofthecuts? ??iA
?
??????? ??iA
p
? whereA
?
?????
A
p
?W? Usingrelation????we ?ndA
?
?? ?as?
i
?x??? forx?A
?
??A
p
?W
?as?
i
?x??? forx?WnA
p
? andp?? ?ifp??wew ould have?
i
?x???
i
?y? for
x?A
p
nA
?
andy?A
?
??
Byrelation ????we can suppose thatS
?
andS
?
areb oth nested?Therefore?
S
i
consistsof thecuts? ??i? and? ??iW? fori??? ?? It followsthat?
?
?xy??
?
?
?xy??
?
?
forx??y?W? Usingrelation ????we obtain??
?
?xy??? for
x??y?W?Therefore? allcuts? ???u?b elong toS foru?W?Together with
? ???? and? ???? theyforma setof ?vepairwisecrossingcuts? which contradicts
theassumptionthatS is??nested?
AsS is nottrivial?the minimality assumptiononSimpliesthat one ofthe ??
splits isnotpresent inS? say?? ??? ??S? LetA
?
?????A
p
denote the?inclusion wise?
minimalsubsetsofVnf?g forwhich??A
?
?f?g????????A
p
?f?g?belongtoS and
set
S
min
??f? ??A
?
??????? ??A
p
?g?
A setT?Vnf?g is saidtobe transversal ifT meets each of thesetsA
?
?????A
p
?
Proposition ???p??andthesetsA
?
?????A
p
arepairwisedisjoint?
Proof?We ?rstclaim that
??? everytransversalT hascardinalityjTj??
Indeed?ifjTj?? then?inview ofProposition ???thereexists??S??Sforwhich
S??T?f?g??f?g?Then?TisdisjointfromthesetA
i
for which?A
i
?S?
??
Page 14
contradicting theassumption thatT is transversal?
If? ??A
i
??? ??A
j
??S
min
aretwocross?freecuts? then thefollo wingholds?
???A
i
?A
j
?? andjA
i
j?jA
j
j???
Indeed?A
i
?A
j
?Vnf?g? ??? ???since? ??A
i
? and? ??A
j
? arecross?free? Moreov er?
jA
i
nA
j
j?? ?else? theset?A
i
nA
j
??fxg wherex?A
j
nA
i
w ouldbea transv ersal
ofcardinality? ??contradicting ???? and?similarly?jA
j
nA
i
j? ??Relation
???now followsfrom the above observations andthe iden tity???jA
i
?A
j
j?
jA
i
nA
j
j?jA
j
nA
i
j?jA
i
?A
j
j?We now showthat
???? ev erytwo cutsamong? ??A
?
??????? ??A
p
?
?
arecrossing?
F or?supp osenot? Then?by ????the cutsareof theform?? ??A
i
??? ??A
?
i
? fori?
??????q and? ??A
j
? forj?q???????m?whereA
?
i
??Vn?A
i
?f?g? andp?m?q?
Clearly?m?? sincethe cuts? ??A
?
??????? ??A
m
? arepairwisecrossing?Weclaim
thatwe can?nda transversalofcardinality ?? th uscontradicting ???and proving
?????F orthis?weuse thefactthatA
i
?A
j
?A
i
?A
?
j
?A
?
j
?A
?
h
??? for??i?m?
?? j?h?q? Indeed let ussupp ose thatq?? ?thecasewhenq?? isanalogue??
Then?by the aboveobserv ation? one of thetwo setsA
?
?A
?
?A
?
andA
?
?
?A
?
?A
?
is
not empty?similarly? oneof thetwo setsA
?
?
?A
?
?
?A andA
?
?
?A
?
?
?A
?
?
isnot empty?
We canassume withoutlossofgeneralitythatA
?
?A
?
?A
?
?A
?
?
?A
?
?
?A
?
?
????
Then?c hoosingx?A
?
?A
?
?A
?
?y?A
?
?
?A
?
?
?A
?
?
andz?A
?
?
?A
?
? theset
fx?y?zg istransv ersal?
We canconclude the proofof Proposition???Indeed?p??by ???and p??
by ????? hence?p? ?? Moreover?the setsA
?
?????A
?
arepairwisedisjoint for?
otherwise?wewould ?ndatransversalofcardinality lessthan ??
We nowconcludethe proof ofTheorem?byanalysing thev ariouspossibilities
forthefamily S
min
? Thefollowingnotationwillbeuseful?Giventwodisjointsets
S andA?S
A
denotesa setoftheformS?BwhereB?A?
We?rstassumethatthefamilyS
min
containsa cut? ??A
i
?withjA
i
j???Then?
wecanassumethatthecutsinS
min
areoftheform
????B
?
??????B
?
??????B
?
???????B
?
?
where B
?
?????B
?
arepairwisedisjointsubsetsoff?g? Let
S?C
?
?C
?
?C
?
?C
?
beadecompositionofSintofournestedfamilieswhere???iA
i
??C
i
for i??????
and?????A
?
??C
?
?
ConsiderthesetX???????Allinduced??splitsonXmustbepresentinS
X
?
therefore?
????
???
??????
???
??S?
??
Page 15
The abovetwo cuts arecrossing? moreov er?they are crossingwith? ???A
?
??ob vi?
ous? andwith? ????A
?
? ?use heretheminimalityassumption on ??A
?
? and? th us?
theym ustbeassigned toC
?
?C
?
? Byconsidering thesetsX ?? ????and?????
we obtainin thesame mannerthat? ???
???
??? ???
???
?b elongtoC
?
?C
?
and that
? ???
???
??? ???
???
?b elongtoC
?
?C
?
? Without lossof generality? let usassign? ???
???
?
toC
?
and? ???
???
? toC
?
? then?necessarily?? ???
???
??C
?
?? ???
???
??C
?
andwe
reacha contradictionwhen tryingto assign? ???
???
? toC
?
?C
?
?
We can now assumethatjA
i
j?? forev erycut? ??A
i
??S
min
?Therefore?S
min
consists ofthecuts
??????? ?????? ?????? ?? ??
and? thus?? ?????????? ??S? Let
S?C
?
?C
?
?C
?
?C
?
beadecomp ositionofS in to fournestedfamilies where? ??i??C
i
fori??????? ??
F or every elementk?V for which??k? ??S?we ?ndsimilarlythatS con tains
four cuts ofthe form??ki??i?Vnfkg??It follows that atleast oneof? ????? ???
b elongstoS? Say?? ????S andwe cansupp osethat
? ????C
?
?
The followingobservationwillberepeatedly used?Any cutb elonging toC
?
and
distinct from? ???isof the form??S?where ???S and? ??S?
F or each ofthe setsX ??????? ????? and????? all ??splitsarepresent inS
X
?
therefore?? ???
???
??? ???
???
??? ???
???
??S? It is easytov erify thatthesecutsm ust
be assignedin thefollo wingmanner totheclassesC
i
composingS?
? ???
???
??C
?
?? ???
???
??C
?
?? ???
???
??C
?
?
Considering thesetX ???????we seethat? ???
???
??S?We can assume that
? ???
???
??C
?
? Then?? ?????? ???
???
??????
???
?arenestedwhichimplies that
? ??????C
?
?
Thisyields? ????S??Indeed? if? ??? ??S?thenS containsfourcuts ofthe form
??i???we reacha contradictionsinceany cut??i?? iscrossingwith?????? and th us
cannotbeassigned toC
?
?? Withoutloss ofgenerality?
? ????C
?
?
ConsideringthesetX???????wederiveanalogouslythat
? ???
???
??C
?
?
??
Page 16
Wewill use the following fact?
????F orX ?? ????? theinduced cutfamilyS
X
contains no ??split?
F or? if not?then? ???
???
??S?yieldinga contradiction as thiscutcannotbe
assignedto any classC
i
?
Inparticular?weobtain that the cut? ???
???
? ?whichb elongs
i
toC
?
? isequal
???
to????????
?
Consideringthe set
???
X ??
?
?????
?
we obtainthat? ???
???
???
?b elongstoS?
Moreov er?
? ???
???
??C
?
?
?Indeed?? ???
???
? ??C
?
?C
?
since itcrosses? ????and? ?????? If? ???
???
??C
?
? then
itis nestedwith? ???
???
? which implies that? ??????S contradicting??????
Considering theset X ?? ?????we obtainthat? ???
???
??S?We now reacha
contradiction sincewe cannotassign thiscutto any classC? Indeed?? ???? ??
C
?
?C ?obviously? and? ???? ??C?C ?for? otherwise?? ???? is nested?either
with???????? orwith? ?????and? ???
???
?? which impliesthatone of the cuts? ??????
? ??????b elongs toS? contradicting ?????? Thisconcludesthe proofof Theorem ??
? Conclusions
We have presented somerelationsb etw eenConjecture??dealing withthe maxi?
m umcardinality of anequilateralset inthek ?dimensionalrectilinearspace? and
some othergeometricquestions? like the maxim um sizea?n? of anan tichain ina
design onnp oints?or the touchingn umb ersof the cross?polytope andthesimplex?
We men tionheresome furtherrelatedproblems?
Considerthe sequence?n?a?n??
n??
? Isitmonotone nondecreasing? Do esit
conv ergeto?? ?Ifthesequencew ouldbeb oundedbya constantC? itw ould
imply the upperb ound?k?C fore??
?
?k????
Itw ouldbe interesting to evaluate thetouc hingn umb ert?P? ofak?dimensional
polytop e?Conjecture?asserts that? forP??
k
?thek?dimensional cross?polytop e??
thisn umb er is equal to?k ?then umb erofv ertices of?
k
?? IfP is thek?dimensional
cube? thent?P???
k
?then umber ofvertices?? Ontheotherhand? forP??
k
?thek?dimensionalsimplex??thisn umb eris?k?? ifk?? ?thus? greater than
then umb er ofvertices?? One mayw onder forwhichp olytopesP? then umb er of
vertices ofP is anupp erbound fort?P ??Is ittrue whenP iscen trallysymmetric
? The answerisobviouslyp ositivewhenthen umb erofv ertices ofPexceeds?
k
which is thecase? forinstance?ifPisak?dimensionalzonotop e?Giv enap olytope
P anditssymmetrization P
?
??P?P? observe thatt?P? isequaltot?P
?
??Hence?
iftheanswertotheabovequestion ispositiv e?we ?nd thatt??
k
??k?k? ???
??
Page 17
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its Applicationsv ol???? Cambridge University Press??????
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