Nowhere differentiable intrinsic Lipschitz graphs

This is a sel -a chi ed e sion o an o iginal a icle. This e sion may di e om he o iginal in pagina ion and ypog aphic de ails. Au ho (s): Ti le: Yea : Ve sion: Copy igh : Righ s: Righ s u l: Please ci e he o iginal e sion: CC BY 4.0 h ps://c ea i ecommons.o g/licenses/by/4.0/ 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.