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Nowhe e di e en iable in insic Lipschi z g aphs
© 2021 The Au ho s. Bulle in o he London Ma hema ical Socie y is copy igh © London Ma hema ical Socie y.
Published e sion
Julia, An oine; Nicolussi Golo, Sebas iano; Vi one, Da ide
Julia, A., Nicolussi Golo, S., & Vi one, D. (2021). Nowhe e di e en iable in insic Lipschi z
g aphs. Bulle in o he London Ma hema ical Socie y, 53(6), 1766-1775.
h ps://doi.o g/10.1112/blms.12540
2021
Bull. London Ma h. Soc. 0 (2021) 1–10 doi:10.1112/blms.12540
Nowhe e diffe en iable in insic Lipschi z g aphs
An oine Julia, Sebas iano Nicolussi Golo and Da ide Vi one
Abs ac
We cons uc in insic Lipschi z g aphs in Ca no g oups wi h he p ope y ha , a e e y poin ,
he e exis infini ely many diffe en blow-up limi s, none o which is a homogeneous subg oup.
This p o ides coun e examples o a Rademache heo em o in insic Lipschi z g aphs.
The no ion o Lipschi z submani olds in sub-Riemannian geome y was in oduced, a leas in
he se ing o Ca no g oups, by F anchi, Se apioni and Se a Cassano in a se ies o seminal
pape s [5–7] h ough he heo y o in insic Lipschi z g aphs. One o he main open ques ions
conce ns he diffe en iabili y p ope ies o such g aphs: in his pape , we p o ide examples o
in insic Lipschi z g aphs o codimension 2 (o highe ) ha a e nowhe e diffe en iable, ha is,
ha admi no homogeneous angen subg oup a any poin .
Recall ha a Ca no g oup Gis a connec ed, simply connec ed and nilpo en Lie g oup
whose Lie algeb a is s a ified, ha is, i can be decomposed as he di ec sum ⊕s
j=1Vjo
subspaces such ha
Vj+1 =[V1,V
j] o e e y j=1,...,s−1,[V1,V
s]={0},V
s={0}.
We shall iden i y he g oup Gwi h i s Lie algeb a ia he exponen ial map exp : ⊕s
j=1Vj→G,
which is a diffeomo phism. In his way, o λ>0, one can in oduce he homogeneous dila ions
δλ:G→Gas he g oup au omo phisms defined by δλ(p)=λjp o e e y p∈Vj. A subg oup
o Gis said o be homogeneous i i is dila ion-in a ian . Assume ha a spli ing G=WV
o Gas he p oduc o homogeneous and complemen a y ( ha is, such ha W∩V={0})
subg oups is fixed; we say ha a unc ion φ:W→Vin insic Lipschi z i he e is an open
nonemp y cone Usuch ha V {0}⊂Uand
pU ∩Γφ=∅ o allp∈Γφ,
whe e Γφ={wφ(w):w∈W}is he in insic g aph o φ. We say ha a se Σ ⊂Gis a blow-up
o Γφa ˆp=ˆwφ(ˆw) i he e exis s a sequence (λn)nsuch ha λn→+∞and he limi
lim
n→∞δλn(ˆp−1Γφ)=Σ
holds wi h espec o he local Hausdo ff con e gence. I is wo h ecalling ha , i φis
in insic Lipschi z, hen e e y blow-up is au oma ically he in insic Lipschi z g aph o a map
W→V. E en ually, we say ha φis in insically diffe en iable a ˆw∈Wi he blow-up o Γφ
a ˆp=ˆwφ(ˆw) is unique and i is a homogeneous subg oup o G. See [8] o de ails.
Recei ed 8 Janua y 2021; e ised 7 Ap il 2021.
2020 Ma hema ics Subjec Classifica ion 53C17 (p ima y), 22E25, 58C20 (seconda y).
AJ has been suppo ed by he Simons Founda ion Wa e P ojec . SNG has been suppo ed by he Academy o
Finland (g an 322898 ‘Sub-Riemannian Geome y ia Me ic-geome y and Lie-g oup Theo y’). DV has been
suppo ed by FFABR 2017 o MIUR (I aly) and by GNAMPA o INdAM (I aly). All h ee au ho s ha e been
suppo ed by he Uni e si y o Pado a STARS P ojec ‘Sub-Riemannian Geome y and Geome ic Measu e
Theo y Issues: Old and New’.
C
e2021 The Au ho s. Bulle in o he London Ma hema ical Socie y is copy igh C
eLondon Ma hema ical
Socie y. This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which
pe mi s use, dis ibu ion and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
2ANTOINE JULIA, NICOLUSSI GOLO AND DAVIDE VITTONE
We say ha a g oup Galong wi h a spli ing WV sa isfies an in insic Rademache Theo em
i all in insic Lipschi z maps φ:W→Va e in insically diffe en iable almos e e ywhe e ( ha
is, o almos all poin s o Wequipped wi h i s Haa measu e). I was p o ed in [6] ha his
is he case when V≃Rand Gis o s ep wo; o he pa ial esul s o g aphs wi h codimension
1(V≃R) a e con ained in [4, 9]. I Vis a no mal subg oup, he Rademache Theo em has
been p o ed o gene al Gby An onelli and Me lo in [2]. Recen ly, he hi d-named au ho [12]
p o ed ha Heisenbe g g oups (wi h any spli ing) sa is y an in insic Rademache Theo em.
The ques ion has been open o a long ime i Gis he Engel g oup (which has s ep 3) and
V≃R(see [1]). In his pape , we p o e a esul in he nega i e di ec ion: namely, we p o ide
examples o in insic Lipschi z g aphs ha a e nowhe e in insically diffe en iable. Le us s a e
ou main esul :
Theo em 1. Le Gbe a Ca no g oup wi h s a ifica ion s
j=1 Vj.Le WV be a
spli ing o Gsuch ha W∩V2⊂ [W,W]and he e exis s 0∈V∩V1such ha 0=0
and [ 0,W]=0. Then he e is an in insic Lipschi z unc ion φ:W→V ha is nowhe e
in insically diffe en iable.
Mo eo e , φcan be cons uc ed in such a way ha , o e e y p∈Γφ, he ollowing p ope ies
hold.
(a) The e exis infini ely many diffe en blow-ups o Γφa p.
(b) No blow-up o Γφa pis a homogeneous subg oup.
The p oo o Theo em 1is pos poned in o de o fi s p o ide some commen s.
Rema k 1. The simples example o a Ca no g oup whe e Theo em 1applies is G=H×R,
whe e His he fi s Heisenbe g g oup. As cus oma y, we conside gene a o s X, Y,T o he
Lie algeb a o Hsuch ha [X, Y ]=T,[X,T]=[Y,T] = 0 and fix he exponen ial coo dina es
(x, y, )=exp(xX +yY + T ). Using coo dina es (x, y, , )onH×Rwi h ∈R,wecan
conside he spli ing H×R=WV gi en by he e ical subg oup W={x= =0}o H
and he ho izon al Abelian subg oup V={y= =0}. Then V2∩W⊂ [W,W]={0}and
0=(0,0,0,1) commu es wi h W. Hence, his spli ing o H×Rsa isfies he condi ions o
Theo em 1and i does no sa is y an in insic Rademache Theo em.
I is wo h obse ing ha , in his se ing, he map φ:W→Vp o ided in he p oo o
Theo em 1 akes he o m φ(y, )=(0,u( )), whe e uis he 1
2-H¨olde con inuous unc ion
cons uc ed in he Appendix. In pa icula , he in insic g aph Γφis he se {(0,y, ,u( )) :
y, ∈R}and i is con ained in he Abelian subg oup W×R. One o he p ope ies o uis ha
he limi
lim
s→ |u( )−u(s)|
| −s|
does no exis s a any ∈Rand his is he ul ima e eason o he nondiffe en iabili y o φ.
Simila coun e examples can be cons uc ed in any codimension k⩾2: in ac one can
conside Hk−1×R=(Rk−1
x×Rk−1
y×R )×R wi h spli ing WV defined by W={x=0, =
0},V={y=0, =0}. I can be easily checked ha he map φ(y, )=(0,u( )) defines an
in insic Lipschi z g aph o codimension k o which he p ope ies (a) and (b) in Theo em 1
hold a e e y poin .
Rema k 2. The measu e μ=HdΓφ, whe e dis he Hausdo ff dimension o Wand Hdis
he d-dimensional Hausdo ff measu e, does no ha e a unique angen measu e a any poin .
Indeed, fi s , any angen measu e o μis suppo ed on a blow-up o Γφ. Second, by [7, Theo em
3.9], μand all i s dila ions a e uni o mly d-Ahl o s egula , and hus any angen measu e o μis
NOWHERE DIFFERENTIABLE INTRINSIC LIPSCHITZ GRAPHS 3
d-Ahl o s egula . We hen conclude ha i μ1and μ2a e wo angen measu es o μsuppo ed
on diffe en blow-ups o Γφ, hen hey a e wo dis inc measu es. Since blow-ups o Γφa e no
unique, so a e angen measu es. Obse e also ha no angen measu e can be fla , ha is,
suppo ed on a homogeneous subg oup. In pa icula , Γφis pu ely C1
H-un ec ifiable, ha is,
Hd(Γφ∩Σ) = 0 o e e y submani old Σ o class C1
H(see, o example, [3,§2.5 and 6.1]).
Rema k 3. I Wis a homogeneous subg oup o Gwi h codimension 1, hen he condi ions
o Theo em 1canno be me because s
j=2 Vj=[W,W]+[W,V]. Ac ually, in insic Lipschi z
g aphs o codimension 1 a e bounda ies o se s wi h fini e pe ime e in G(see, o example, [11,
Theo em 1.2]), hence a almos e e y poin hey possess a leas one blow-up which is a
homogeneous subg oup o codimension 1, see [1]. The e o e, any possible coun e example o
he Rademache Theo em in codimension 1 canno be as s iking as he one p o ided by
Theo em 1, in he sense ha p ope y (b) canno hold on a se wi h posi i e measu e.
Rema k 4. Following he same p oo s a egy, one can ex end Theo em 1 o he case
W∩Vj⊂ [W,W] o some j>2and 0∈Vk∩V {0}wi h k<jand [ 0,W] = 0, by aking
ak/j-H¨olde analogue o he unc ion ucons uc ed in he appendix.
P oo o Theo em 1. Le β:W→Rbe a nonze o linea unc ion such ha W∩Vj⊂ke β
whene e j=2and[W,W]⊂ke β;suchaβexis s†because W∩V2⊂ [W,W]. No e ha such
a unc ion βis in ac a g oup mo phism W→R.
Conside a 1/2-H¨olde con inuous unc ion u:R→Rwi h he ollowing p ope ies. Fi s ,
he diffe ence quo ien s
Δ(s, )= u(s)−u( )
sgn(s− )|s− |1/2
a e bounded, namely,
|Δ(s, )|⩽1 o e e ys, ∈R.(1)
Second, he e exis c1>0andc2>0 such ha , o e e y 0∈Rand δ∈(0,1 ], he e exis
s1,s
2∈Rsuch ha
sgn(s1− 0)=sgn(s2− 0)
c1δ⩽|s1− 0|⩽δ
c1δ⩽|s2− 0|⩽δ
|Δ(s1,
0)−Δ(s2,
0)|⩾c2.
(2)
Such a unc ion exis s, as we show in he Appendix.
We can hen define φ:W→Vas
φ(w)=u(β(w)) 0.
No e ha he condi ion [ 0,W] = 0 implies
w =w o allw∈Wand ∈R 0.(3)
The e o e, by he Bake –Campbell–Hausdo ff o mula, he in insic g aph o φis he se o
poin s wφ(w)=w+u(β(w)) 0 o w∈W.
†Fo ins ance, one can conside β(x)=x, w0 o some w0∈(W∩V2) [W,W] and a scala p oduc on W
adap ed o he g ading s
j=1 W∩Vjo W.
4ANTOINE JULIA, NICOLUSSI GOLO AND DAVIDE VITTONE
Claim 1. The map φis in insic Lipschi z.
Fix a homogeneous no m ·on G. No e ha , since β(δλx)=λ2β(x) o all x∈W, he e is
a cons an Csuch ha |β(x)|⩽Cx2, o all x∈W. We check ha Γφhas he cone p ope y
o he cone (see [7, Defini ion 10])
U={w :w∈W, ∈V, >2√C 0w}.
Gi en ˆw,w ∈W,by(3)weha e(ˆwφ(ˆw))−1(wφ(w)) = ( ˆw−1w)(φ(ˆw)−1φ(w)) and
φ(ˆw)−1φ(w)=|u(β(w)) −u(β(ˆw))| 0⩽|β(w)−β(ˆw)|1/2 0
=|β(ˆw−1w)|1/2 0⩽√Cˆw−1w 0.
Thus, ( ˆwφ(ˆw))−1Γφ∩U=∅ o all ˆw∈W, ha is,Γ
φis an in insic Lipschi z g aph.
Claim 2. Fo p∈Γφ, none o he blow-ups o Γφa pis a homogeneous subg oup.
We fi s obse e ha , i V0⊂V∩V1is he ho izon al subg oup gene a ed by 0and L:W→
V0pa ame e izes a homogeneous subg oup ΓLo G, hen L|W∩V2= 0. Indeed, he homogenei y
o ΓLimplies ha o e e y w∈W∩V2one has L(2w)=√2L(w), because
(2w)(√2L(w)) = δ√2(w)δ√2(L(w)) = δ√2(wL(w)) ∈ΓL,
while he ac ha ΓLis a subg oup (plus he ac ha V0and Wcommu e) gi es L(2w)=
2L(w), because
wwL(w)L(w)=(wL(w))(wL(w)) ∈ΓL.
This p o es ha L=0onW∩V2.
We now p o e he claim. Assume by con adic ion ha he e exis ˆp=ˆwφ(ˆw)∈Γφ, a map
L:W→Vsuch ha he in insic g aph ΓLo Lis a homogeneous subg oup and a sequence
(λn)nwi h λn→+∞,and
lim
n→∞δλn(ˆp−1Γφ)=Γ
L.
Obse e ha o e e y w∈Wand e e y n
δλn(( ˆwφ(ˆw))−1(wφ(w))) = δλn(ˆw−1wφ(ˆw)−1φ(w))
=δλn(ˆw−1w)u(β(w)) −u(β(ˆw))
1/λn
0.
I we se w=ˆwδ1/λnw, hen β(w)=β(ˆw)+β(w)/λ2
n. The e o e, he se δλn(ˆp−1Γφ)is he
in insic g aph o he unc ion om W o Vgi en by
φˆp,λn(w)=u(β(ˆw)+β(w)/λ2
n)−u(β(ˆw))
1/λn
0.
Since he maps φˆp,λn ake alues in V0,Lis also V0- alued and, as we saw abo e, his implies
ha L|W∩V2=0.
W i e ˆ
=β(ˆw) and le w0∈W∩V2be such ha β(w0) = 1; hen o e e y h∈R
φˆp,λn(hw0) = (sgn h)|h|1/2Δ(ˆ
+h/λ2
n,ˆ
) 0.(4)
By (2), he e exis s a sequence (hn)nsuch ha o e e y n
|hn|∈[c1,1] and φˆp,λn(hnw0)⩾√c1c2 0/2.

NOWHERE DIFFERENTIABLE INTRINSIC LIPSCHITZ GRAPHS 5
Up o passing o a subsequence we can also assume ha hn→¯
hwi h |¯
h|∈[c1,1 ]; since
φˆp,λn(hnw0)−φˆp,λn(¯
hw0)=
u(ˆ
+hn/λ2
n)−u(ˆ
+¯
h/λ2
n)
1/λn 0,
⩽|hn−¯
h|1/2 0
we ob ain
L(¯
hw0)= lim
nφˆp,λn(¯
hw0)= lim
nφˆp,λn(hnw0)⩾√c1c2 0/2.
This con adic s he ac ha L(¯
hw0) = 0, and he claim is p o ed.
Claim 3. Fo p∈Γφ, he e exis infini ely many diffe en blow-ups o Γφa p.
Le ˆp=ˆwφ(ˆw)∈Γφbe fixed and le ˆ
=β(ˆw); as be o e, fix also w0∈W∩V2such ha
β(w0)=1.By(2), we can find infini esimal sequences (s1
n)n,(s2
n)nsuch ha
sgn(s1
n)=sgn(s2
n) o e e y n,
Δ(ˆ
+s1
n,ˆ
)⩾Δ(ˆ
+s2
n,ˆ
)+c2.
Up o passing o a subsequence, we can assume ha he e exis s σ∈{1,−1}and Δ1,Δ2∈R
such ha
sgn(s1
n)=sgn(s2
n)=σ o e e yn,
Δ(ˆ
+s1
n,ˆ
)→Δ1and Δ(ˆ
+s2
n,ˆ
)→Δ2as n→∞,
Δ1⩾Δ2+c2.
Due o he con inui y o s→ Δ(ˆ
+s, ˆ
) o s= 0, gi en Δ ∈(Δ2,Δ1) one can find an
infini esimal sequence (sn)nsuch ha , o e e y n,sgn(sn)=σand Δ(ˆ
+sn,ˆ
) = Δ. Now, as
in (4) he se δ|sn|−1/2(ˆp−1Γφ) is he in insic g aph o a map φˆp,|sn|−1/2:W→Vsuch ha
φˆp,|sn|−1/2(σw0)=σΔ(ˆ
+sn,ˆ
) 0=σΔ 0.
Since he amily (φˆp,|sn|−1/2)nis uni o mly H¨olde con inuous, up o ex ac ing a subsequence i
con e ges locally uni o mly o a map ψ:W→Vsuch ha ψ(σw0)=σΔ 0. The a bi a iness o
Δ∈(Δ2,Δ1) implies ha he e a e infini ely many diffe en blow-ups a ˆp, and his concludes
he p oo . 
Appendix
We a e now going o cons uc he unc ion uused in he p oo o Theo em 1: his unc ion, in a
sense, p o ides a coun e -example o a Rademache p ope y o Lipschi z unc ions om (R,|·
|1/2) o(R,|·|). We will use a classical p ocedu e p oducing a sel -simila unc ion: al hough
hese ideas a e well-known (see, o example, [10] and he e e ences he ein), we p e e o
include a de ailed cons uc ion because we we e no able o find in he li e a u e explici
s a emen s o he p ecise es ima es (2) we need.
We cons uc a unc ion u:[0,1]→[0,1 ] whose diffe ence quo ien s
Δ(s, )= u(s)−u( )
sgn(s− )|s− |1/2
sa is y
|Δ(s, )|⩽1 o e e ys, ∈[0,1].(A.1)
We will cons uc uin such a way ha he e exis c1>0andc2>0 wi h he p ope y ha ,
o e e y ∈[0,1] and δ∈(0,1 ], one can find s1,s
2∈[0,1 ] such ha he condi ions in (2)
6ANTOINE JULIA, NICOLUSSI GOLO AND DAVIDE VITTONE
Figu e A.1 (colou online).Fou ins ances o he unc ions undefined in (A.2).
hold. One can hen ex end u o Rby se ing u( )=u(− ) o ∈[−1,0]and u( +2n)=u( )
o all n∈Z: his ex ended udoes sa is y (1) and (2).
The unc ion uis ob ained as he limi o a sequence (un)n∈Nwhe e u0( )= . The unc ion
un+1 is ob ained om unon se ing
un+1( )=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
2
3un9
4 i ∈0,4
9,
2
3−1
3un9 −4
9 i ∈4
9,5
9,
1
3+2
3un9
4 −5
9 i ∈5
9,1.
(A.2)
The fi s ew o he unc ions u0,u
1,u
2,... a e plo ed in Figu e A.1. Le us no e ha un(0) = 0
and un(1) = 1 o e e y n, hence un(4/9) = 2/3andun(5/9) = 1/3 o e e y n⩾1.
No e (see Figu e A.2) ha he g aph o un+1 is he union o h ee affine copies o he g aph
o un, ia he ollowing maps (ac ing on p∈R2):
A0(p)=4/90
02/3p,
A4/9(p)=1/90
0−1/3p+4/9
2/3,
A5/9(p)=4/90
02/3p+5/9
1/3.
(A.3)
Claim 1. The unc ions uncon e ge uni o mly on [ 0,1 ] o a unc ion u o which (A.1) holds.
The ac ha ununi o mly con e ge o a con inuous unc ion uis a consequence o he
es ima e
un+1 −unC0([ 0,1]) ⩽2
3un−un−1C0([ 0,1]).
NOWHERE DIFFERENTIABLE INTRINSIC LIPSCHITZ GRAPHS 7
Figu e A.2 (colou online).I e a ed images o he uni squa e unde he affine maps in (A.3);
do s a e he images o (0,0) and (1,1), and hey belong o he g aph o he limi unc ion u
This es ima e ollows di ec ly om he defini ion (A.2): o ins ance, o ∈[0,4/9 ], one has
|un+1( )−un( )|=2
3|un(9 /4) −un−1(9 /4)|⩽2
3un−un−1C0([ 0,1]).
Simila ly, one can ea he o he wo cases ∈[4/9,5/9] and ∈[5/9,1].
The bound (A.1) on he diffe ence quo ien s o u ollows om he ac ha he same is ue
o all unin he sequence, as we a e now going o p o e by induc ion on n. The s a emen is
clea ly ue o n= 0. Suppose ha unsa isfies
|un( )−un(s)|⩽| −s|1/2 o e e ys, ∈[0,1],
we will p o e ha also |un+1( )−un+1(s)|⩽| −s|1/2 o e e y s, ∈[0,1 ]. We dis inguish
se e al cases depending on which in e als ([ 0,4/9], [4/9,5/9] o [5/9,1 ]) he poin s sand
belong o. We can suppose ha s< .
Case 1: sand a e in he same in e al. We can use (A.2) and he induc ion hypo hesis
o conclude.
Case 2: s∈[0,4/9]and ∈[4/9,5/9 ]. Since 0 ⩽un⩽1, one sees om he defini ion o un+1
ha max(un+1(s),u
n+1( )) ⩽2/3=un+1(4/9). Thus
|un+1( )−un+1(s)|⩽max(un+1(4/9) −un+1( ),u
n+1(4/9) −un+1(s))
⩽max(( −4/9)1/2,(4/9−s)1/2)⩽( −s)1/2,
whe e he second inequali y ollows om Case 1.
Case 3: s∈[4/9,5/9] and ∈[5/9,1 ]. Due o he symme y un(x)=1−un(1 −x), his is
simila o Case 2.
Case 4: s∈[0,4/9]and ∈[5/9,1 ]. Then ei he |un+1( )−un+1(s)|⩽1/3, and we a e done
because | −s|⩾1/9, o |un+1( )−un+1(s)|>1/3, and hen necessa ily un+1(s)<u
n+1( )
(o he wise, 0 ⩽un+1(s)−un+1( )⩽un+1(4/9) −un+1(5/9) = 2/3−1/3=1/3) and
|un+1( )−un+1(s)|=un+1( )−un+1(s)
=un+1( )−un+1(5/9) −1/3+un+1(4/9) −un+1(s)
⩽( −5/9)1/2−1/3+(4/9−s)1/2,
8ANTOINE JULIA, NICOLUSSI GOLO AND DAVIDE VITTONE
whe e in he las inequali y we used Case 1. By squa ing he igh -hand side o he las
inequali y, we ob ain
( −5/9)1/2−1/3+(4/9−s)1/22
=( −s)+2( −5/9)1/2(4/9−s)1/2−2
3( −5/9)1/2−2
3(4/9−s)1/2
=( −s)+( −5/9)1/2(4/9−s)1/2−2/3+(4/9−s)1/2( −5/9)1/2−2/3
⩽ −s,
whe e we used he ac ha 4/9−s⩽4/9and −5/9⩽4/9. This is enough o conclude.
Claim 2. The e exis d1>0andd2>0 such ha , o e e y 0∈[0,1 ], one can find
s1,s
2∈[0,1 ] such ha
sgn(s1− 0)=sgn(s2− 0)
d1⩽|s1− 0|⩽1
d1⩽|s2− 0|⩽1
|Δ(s1,
0)−Δ(s2,
0)|⩾d2.
(A.4)
In ac , we will p o e Claim 2 o d1=1/18 and
d2= min 1
31−4
81 −1/2−1,7
9−3
5,1
√5.
We dis inguish se e al cases.
Case 1: 0∈[0,4/9 ]. In his case, i suffices o conside s1=5/9ands2=1, as we now
show. Obse e ha he dis ances o s1,s
2 om 0a e bo h g ea e han 1/9>d
1.
I u( 0)⩾2/9, by (A.1) and he equali y u(0) = 0, we ha e 1/2
0⩾u( 0), hence 0⩾4/81;
since u( 0)⩽2/3, we ob ain
Δ(1,
0)⩾
1
3
1−4
81
and Δ(5/9,
0)⩽
1
3−2
9
1
9
=1
3,
so ha Δ(1,
0)−Δ(5/9,
0)⩾d2.
I u( 0)⩽2/9, hen (4/9− 0)1/2⩾2/3−2/9=4/9, hence 5/9− 0⩾1/9+(4/9)2=
(5/9)2and
Δ(1,
0)⩾7
9and Δ(5/9,
0)⩽
1
3
5
9
=3
5
and again Δ(1,
0)−Δ(5/9,
0)⩾d2.
Case 2: 0∈[4/9,1/2 ]. In his case, we ake s1=5/9ands2= 1. The dis ances o s1,s
2
om 0a e bo h no less han 1/18 = d1and, since 1/3⩽u( 0)⩽2/3, one ge s
Δ(1,
0)⩾
1
3
5
9
=1
√5and Δ(5/9,
0)⩽0.
Case 3: 0∈[1/2,1 ]. We p o ed ha , i 0∈[0,1/2 ], he claim can be p o ed on choosing
s1=5/9ands2= 1. The e o e, due o he symme y u(x)=1−u(1 −x), when 0∈[1/2,1]
i is enough o ake s1=0ands2=4/9.
Claim 3. The e exis c1>0andc2>0 such ha , o e e y ∈[0,1] and δ∈(0,1], one
can find s1,s
2∈[0,1 ] o which he condi ions in (2) hold.