Minimal extension for the α-Manhattan norm

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/ Minimal ex ension o he α-Manha an no m © 2024 Accademia Nazionale dei Lincei Published e sion Campbell, Daniel; Kau anen, Aapo; Radici, Emanuela Campbell, D., Kau anen, A., & Radici, E. (2023). Minimal ex ension o he α-Manha an no m. Rendicon i Lincei: Ma ema ica e Applicazioni, 34(4), 773-807. h ps://doi.o g/10.4171/ lm/1027 2023 Rend. Lincei Ma . Appl. 34 (2023), 773–807 DOI 10.4171/RLM/1027 ©2024 Accademia Nazionale dei Lincei Published by EMS P ess This wo k licensed unde a CC BY 4.0 license Calculus o Va ia ions. – Minimal ex ension o he ˛ -Manha an no m, by Daniel Campbell, Aapo Kau anen and Emanuela Radici, communica ed on 10 No embe 2023. Abs ac . – Le @Q be he bounda y o a con ex polygon in R2 , e˛D.cos ˛; sin ˛/ and e? ˛D.sin ˛; cos ˛/ a basis o R2 o some ˛2Œ0; 2/ and 'W@Q!R2 a con inuous, ini ely piecewise linea injec i e map. We cons uc a ini ely piecewise a ine homeomo phism WQ!R2 coinciding wi h ' on @Q such ha he ollowing p ope y holds: jhD ; e˛ij.Q/ ( esp., hD ; e? ˛ij.Q/ ) is as close as we wan o in jhDu; e˛ij.Q/ ( esp., in jhDu; e? ˛ij.Q/ ) whe e he in imum is mean o e he class o all BV homeomo phisms u ex ending ' inside Q . This esul ex ends ha al eady p o en by P a elli and he hi d au ho in [A i Accad. Naz. Lincei Rend. Lincei Ma . Appl. 29 (2018), no. 3, 511–555] in he shape o he domain. Keywo ds. – Homeomo phic ex ension, BV homeomo phisms, s ic app oxima ion in BV. Ma hema ics Subjec Classi ica ion 2020. – 46E35 (p ima y); 30E10, 58E20 (seconda y). 1. In oduc ion In his pape , we a e in e es ed in he p oblem o ex ending injec i e con inuous and piecewise linea bounda y alues om a con ex polygon by piecewise a ine homeomo phisms. The mo i a ion o such a s udy a ises in he con ex o app oxima ion p oblems ound in egula i y heo y o non-linea elas ici y. The e is al eady a plu ali y o ex ension esul s in a a ie y o con ex s, which ha e been applied o sol e a ious app oxima ion p oblems. Le us now gi e an o e iew o some examples. In gene al, we a e in e es ed in he app oxima ion o a weakly di e en iable homeo- mo phism, which we would like o app oxima e by C1 homeomo phisms o by locally ini e piecewise a ine homeomo phisms. The app oxima ion o a plana W1;p home- omo phism 1 < p < 1 in [9,10] elies hea ily on he injec i i y o he ha monic ex ension o con ex bounda y alues. In [5], he au ho s we e also able o app oxima e a bi-Lipschi z map and i s in e se simul aneously in he .p; p/ -bi-Sobole se ing and o do so used he ex ension esul in [6]. In o de o sol e he W1;1 case in [8], he au ho s had o de elop an independen ex ension esul in ha pape which was u he examined and imp o ed in [1,15]. The ex ension esul was also u ilized in d. campbell, a. kau anen and e. adici 774 he .1; 1/ -bi-Sobole se ing in [12]. Finally, le us men ion ha he au ho s o [14] app oxima e plana BV homeomo phisms using an ex ension esul hey p o ed in [13]. Mo e han jus he app oxima ion o weakly di e en iable homeomo phisms by di eomo phisms, hese ex ension esul s ha e been key in examining he beha io o weak and s ong limi s o homeomo phisms in hei espec i e classes. Such esul s include a ca ego iza ion o he closu e o Hom W1;p , p2 , in [11], a ca ego iza ion o he closu e o Hom W1;p,1<p<2, in [7] and pa ial BV esul in [2,4]. I was demons a ed in [13] ha hei main ex ension esul can be “ o a ed” o app oxima e a BV homeomo phism s ic ly and simila ly in [14] o he a ea-s ic case. Ne e heless, his app oach makes he applica ion o he ex ension esul somewha cumbe some and echnical. The main esul o he p esen pape is a piecewise a ine homeomo phic ex ension ha imp o es on ha o [13]. Mo e p ecisely, we conside ex ensions o piecewise linea bounda y alues de ined on a bounda y o con ex quad ila e als (and no only ec angles pa allel o he coo dina e axes as in [13]) which a e op imal in a pa icula BV sense. We emphasize ha he gene ali y o he class o con ex quad ila e als includes he “ o a ed” e sion o he ex ension esul o [13]. Also, ou Theo em 1.2 is s onge han he ex ension heo em he e (no only because o he shape o Q ) in he sense ha i immedia ely implies hei ex ension heo em bu he opposi e is no ue (see Rema k 1.3). Ne e heless, his imp o emen is a case o sepa a ing es ima es al eady conduc ed in [13]. The mo i a ion o ou ex ension heo em is he ull ca ego iza ion esul in [3], whe e we iden i y a condi ion which gua an ees ha a map is a s ic o a ea-s ic limi o BV homeomo phisms. In he cou se o he app oxima ion, we wan o wo k on g ids ha a e no only made up o ec angles, and we p e e o no ha e o o a e he ec angles. In ha sense, we need he cu en esul , which we p esen below, a e we se some necessa y no a ion. Le QR2 be a con ex polygon, le ˛2Œ0;2/ be ixed and call e˛D.cos˛;sin˛/ and e? ˛D.sin ˛; cos ˛/. We de ine he ollowing numbe s (see Figu e 1): (1.1) aWD in ®hx; e? ˛i W x2Q¯aCWD sup ®hx; e? ˛i W x2Q¯; bWD in ®hx; e˛i W x2Q¯; bCWD sup ®hx; e˛i W x2Q¯: Fo each s2.a; aC/, we de ine V1 s; V 2 suniquely by he condi ions (1.2) V1 s; V 2 s2@Q;hV1 s; e? ˛iDhV2 s; e? ˛i D s; hV1 s; e˛i<hV2 s; e˛i: Simila ly, o e e y 2.b; bC/, we de ine H1 ; H2 uniquely by (1.3) H1 ; H2 2@Q;hH1 ; e˛iDhH2 ; e˛i D ; hH1 ; e? ˛i<hH2 ; e? ˛i: minimal ex ension o he ˛-manha an no m 775 minimal ex ension o he 𝛼-manha an no m 3 𝑒𝛼 {⟨𝑥, 𝑒𝛼⟩=𝑏+} {⟨𝑥, 𝑒𝛼⟩=𝑎−} {⟨𝑥, 𝑒𝛼⟩=𝑏−} {⟨𝑥, 𝑒𝛼⟩=𝑎+} 𝐻1 𝑡 𝑉1 𝑠 𝐻2 𝑡 𝑉2 𝑠 Q Figu e 1. Polygon Qwi h 𝐻1 𝑡, 𝐻2 𝑡, 𝑉1 𝑠and 𝑉2 𝑠. by 𝜌P(A,B) he geodesic dis ance be ween A and B inside P . We de ine he quan i y Ψ𝛼(𝜑):=∫𝑎+ 𝑎− 𝜌P(𝜑(𝑉1 𝑠), 𝜑(𝑉2 𝑠))𝑑𝑠 +∫𝑏+ 𝑏− 𝜌P(𝜑(𝐻1 𝑡), 𝜑(𝐻2 𝑡))𝑑𝑡. Loosely speaking, he quan i y Ψ𝛼(𝜑) accoun s o he leng h o all he geodesics inside P connec ing pai s o poin s on 𝜑(𝜕Q) whose p eimage in 𝜑 is a pai o poin s in 𝜕Q lying on a line pa allel o ei he 𝛼 o 𝛼⊥ . Fu he o 𝑢∈𝐵𝑉 (Ω,R2) we deno e he 𝛼-Manha an no m o 𝐷𝑢 as ∥·∥𝛼which we de ine as ∥𝐷𝑢∥𝛼(Q) :=|⟨𝐷𝑢, 𝑒𝛼⟩|(Q) + |⟨𝐷𝑢, 𝑒⊥ 𝛼⟩|(Q). The main esul s o he pape is a e he ollwoing. Theo em 1.1.Le 𝛼∈ [0,2𝜋) be ixed, Q ⊂ R2 be a con ex polygon and 𝜑:𝜕Q → R2 be a con inuous piecewise linea injec i e map. Then o e e y 𝜀 > 0 he e exis s a ini ely piecewise a ine homeomo phism 𝑣:Q → R2ex ending 𝜑, such ha (1.4) ∥𝐷𝑣∥𝛼(Q) ≤ Ψ𝛼(𝜑) + 𝜀. Figu e 1. Polygon Qwi h H1 ,H2 ,V1 sand V2 s. Le 'W@Q!R2 be con inuous, injec i e and piecewise linea . We deno e P as he bounded componen o R2n'.@Q/ . Fo e e y pai o poin s A;B2x P , we deno e by P.A;B/ he geodesic dis ance be ween A and B inside x P . We de ine he quan i y ‰˛.'/ WD ZaC a P'.V 1 s/; '.V 2 s/ds CZbC b P'.H1 /; '.H2 /d : Loosely speaking, he quan i y ‰˛.'/ accoun s o he leng h o all he geodesics inside P connec ing pai s o poin s on '.@Q/ whose p eimage in ' is a pai o poin s in @Q lying on a line pa allel o ei he ˛ o ˛? . Fu he , o u2BV.; R2/ , we deno e he ˛-Manha an no m o Du as kk˛which we de ine as kDuk˛.Q/WD ˇˇhDu; e˛iˇˇ.Q/CˇˇhDu; e? ˛iˇˇ.Q/: The main esul s o he pape a e he ollowing. Theo em 1.1.Le ˛2Œ0; 2/ be ixed, QR2 a con ex polygon and 'W@Q!R2 a con inuous piecewise linea injec i e map. Then, o e e y ">0 , he e exis s a ini ely piecewise a ine homeomo phism WQ!R2ex ending ', such ha (1.4) kD k˛.Q/‰˛.'/ C": Theo em 1.2.Le ">0and le be he ex ension om Theo em 1.1. Then, (1.5) ˇˇhD ; e˛iˇˇ.Q/ZbC b P'.H1 /; '.H2 /d C"; ˇˇhD ; e? ˛iˇˇ.Q/ZaC a P'.V 1 s/; '.V 2 s/ds C": d. campbell, a. kau anen and e. adici 776 Rema k 1.3.We obse e ha Theo em 1.1 is s onge han he esul o [13] as i p o ides he almos op imal ex ension wi h espec o any o a ed Manha an no m and no jus o he canonical one (whe e ˛D0). Le us also ema k ha Theo em 1.2 immedia ely implies Theo em 1.1 bu he a gumen used o cons uc is exac ly he same. Also, i is immedia e ha ZbC b P'.H1 /; '.H2 /d in ®ˇˇhDu; e˛iˇˇ.x Q/Wu2Hom BV.x Q;R2/; u D'on @Q¯ and ZaC a P'.V 1 s/; '.V 2 s/ds in ®ˇˇhDu; e? ˛iˇˇ.x Q/Wu2Hom BV.x Q;R2/; u D'on @Q¯; and ou esul in ac shows ha he e is a sequence o homeomo phisms achie ing he in imum and ha ing a ia ion con e ging o he le -hand side in he sense o (1.5) . This ac is ac ually a di ec consequence o he p oo s in [13], hough i was no explici ly ema ked he e. The key a gumen is in Theo em 2.9. 1.1. Ske ch o he p oo Be o e expounding he p oo in de ail, le us look a an o e iew o he p oo . We s a wi h a con ex polygon Q . Up o a o a ion o ˛ , we may assume ha ˛D0 . Ei he ( he o a ed) Q has ho izon al sides, o a e emo ing a iny iangle called T1 close o he lowes poin o Q and a iangle called T2 close o he highes poin o Q we ge a con ex  ha has a pai o ho izon al sides (see Figu e 2). We ex end ' on @T1; @T2 so ha i is con inuous injec i e and piecewise linea . By making he iangles small enough, we gua an ee ha ‰0.'.// C‰0.'.T1// C‰0.'.T2// ‰0.'.Q// C" . He e, ou new 'ex ends he o iginal ' om @Q. This s ep is Lemma 3.2. Now, we sepa a e  in o hin ho izon al s ips Si (see Figu e 3), de ining a con inuous injec i e piecewise linea ' on @Si so ha PM iD1‰0.'.@Si// ‰0.'.@// C" which ex ends he o iginal ' om @ [@T1[@T2. This s ep is Lemma 3.3. We sepa a e each Si in o a cen al ec angle and a pai o igh -angle iangles a each end. On he ec angula domains Ri , we can use P oposi ion 2.8 o ex end he bounda y alues and ge a piecewise a ine homeomo phism wi on he Ri sa is ying an es ima e on jDwij.Ri/ . In Lemma 4.2, we show how we ex end he bounda y alues o ge a piecewise a ine homeomo phism on he iangles a he ends o he s ips; see Figu e 6. We do his by u he sepa a ing hem in o e en hinne ec angles whe e minimal ex ension o he ˛-manha an no m 777 we can ex end and es ima e as abo e. The emaining pa o he se is so small ha i s con ibu ion o he no m is bounded by 2i". The inal pa o he p oo is colla ing he es ima es and summing o es ima e ha ou mapping sa is ies (1.4). 2. P elimina ies In his sec ion, we ecall a lis o de ini ions and known geome ical esul s which a e al eady a ailable in he li e a u e. Mos o hem a e aken om [13,14]. No a ion 2.1.Th oughou he pape , we endea o o keep o he ollowing no ms o no a ion: •Qis a con ex polygon, •P is a 2-dimensional polygon wi h bounda y @P . I 'W@Q!R2 is injec i e and piecewise linea con inuous, hen P is he polygon co esponding o he bounded componen o R2n'.@Q/and PD'.@Q/, •˛2Œ0; 2/ is a gi en angle and he ec o e˛WD .cos ˛; sin ˛/ . Also, we deno e e? ˛WD .cos.˛ C=2/; sin.˛ C=2//, •uand a e plana BV mappings, •a;aC; b; bC a e he numbe s om (1.1) , ypically s2.a;aC/ and 2.b; bC/ and `DaCa,hDbCb, •is a con ex polygon wi h 2 sides pa allel o ˛, •T; T1; T2;z T ; T 1 i; T 2 ia e all iangles, •V1 s; V 2 s; H1 ; H2 a e he poin s sa is ying he condi ions in (1.2) and (1.3) al hough we may eplace Qwi h ano he con ex polygon, o example, o T, • by RQDŒa; aCe˛CŒb; bCe? ˛ we deno e he smalles ec angle wi h sides pa allel o e˛and e? ˛con aining Q, •c1; c2; d1; d22Ra e o dina es, • by z C we deno e a gene ic cons an whose p ecise alue may a y be ween es ima es, • he poin s in he p eimage a e A; B; C; D; E; F; G; P; Q,1 • i ' is a piecewise linea injec i e map de ined on he iangle ABC , we call dWD j'.A/ '.C /j he leng h o he image o he hypo enuse h ough ', (1) We do no need o u ilize he no a ion B.x; / D ¹yW jyxj< º so he e is no dange o con usion when using B o deno e a poin . d. campbell, a. kau anen and e. adici 778 •ˇ2.0;  2/is he angle a Ain he iangle ABC , •>0is a small chosen pa ame e , • he poin s in he image a e w i en in bold on , e.g., A;B;C;D;X;Y;Z, •P;P0;PC a e polygons in he image, ypically he piecewise a ine image o a polygon in he p eimage, e.g., PD'.@Q/, •' , a e con inuous injec i e piecewise linea maps om one-dimensional “skele- ons” (i.e., a ini e union o segmen s) in he p eimage, • gi en a se AR2 and a unc ion 'WA!R2 , we deno e by 'eB he es ic ion o ' o a subse BA, •A;B is he geodesic cu e om A o B in P and P.A;B/ is he leng h o ha cu e. De ini ion 2.2 (Geodesics and modi ied geodesics).Le PR2 be a polygon, and le A and B be any wo dis inc poin s in P . We de ine AB as he unique geodesic (i.e., cu e o minimal leng h) connec ing hem, lying inside P . No ice ha AB is a piecewise linea cu e, whose e ices a e only A;B and some e ices o @P whose in e nal angles ha e size a leas  . Assume now ha A;B2@P , and le W1;W2; : : : ; WK be all he e ices o Pme by AB, so ha AB DAW1;W2;:::;WKB: Fix now any ı > 0 . Fo e e y 1iK , le z Wi¤Wi be some a bi a y poin in he in e nal bisec o o he angle a Wi ha ing dis ance om Wi smalle han ı . The piece- wise linea cu e zAB DAz W1;z W2;:::; z WKBis hen called a ı-modi ica ion o AB. No ice ha he e exis s a cons an x ı.P/>0 , depending on P bu no on A and B , such ha he in e io o zAB is con ained in he in e io o P i ı < x ı.P/ , unless he segmen AB is al eady con ained in @P , in which case KD0 and zAB DAB @P . Lemma 2.3 ([13, Lemma 2.4]).Le A;B;C and D be ou dis inc poin s in a polygon P . Then, he in e sec ion AB CD is ei he emp y o closed and connec ed. Assume now also ha A;B;C;D2@P and call @P1; @P2 he wo componen s o @Pn ¹C;Dº . I A2@P1 and B2@P2 , hen AB CD ¤ ; . I A;B2@P1 and AB CD ¤ ; , hen he i s and las poin o his in e sec ion mus ei he be e ices o P o coincide wi h one o he poin s Ao B. We ema k ha in he e e ence he lemma is s a ed wi hou closedness o he in e sec ion bu i ollows om he ollowing simple obse a ion. I A;B;C and D a e ou dis inc poin s in P and he in e sec ion AB CD is no emp y, hen i is ei he a poin (hence a closed se ) o a piecewise linea cu e which s a s and ends a co ne s o @P( hus being he ini e union o closed connec ed segmen s). minimal ex ension o he ˛-manha an no m 779 Lemma 2.4 ([13, Lemma 2.5]).Le P be a polygon, le A;B2@P be wo poin s such ha he segmen AB is no con ained in @P , hen le ı < x ı.P/ and le zAB be a modi ied geodesic in he sense o De ini ion 2.2. Le also P1 and P2 be he wo polygons in which P is di ided by zAB , and le ">0 be a gi en cons an . I ı is small enough, depending only on "and P, hen he ollowing is ue. Fo any wo poin s C;D2Pi o i2 ¹1; 2º, one has (2.1) Pi.C;D/<P.C;D/C": I C2P1 , D2P2 and E2@P1 @P2 is any poin wi h dis ance a mos ı om CD, hen (2.2) P1.C;E/CP2.E;D/<P.C;D/C": De ini ion 2.5 (Se o e ices o a geodesic cu e).Le PR2 be a polygon. Fo e e y A;B2@P , he e is a unique o de ed se X.A;B/D ¹X1; : : : ; XNº such ha he geodesic AB is exac ly he piecewise linea cu e AX1;:::;XNB , and he poin s Xj a e all he e ices o P me by he geodesic AB (excep A and B hemsel es, in case hey a e al eady e ices). The se X.A;B/is called he se o e ices o AB. De ini ion 2.6 ( ı -linea iza ion o a Jo dan cu e).Le be a Jo dan cu e wi h ini e leng h, and le ı > 0 be much smalle han he diame e o he bounded componen o R2n . Le _ A1B1;_ A2B2;:::; _ ANBN be ini ely many essen ially disjoin a cs con ained in . Le hen ' be he closed cu e ob ained by eplacing each a c _ AiBi wi h he segmen AiBi. We say ha 'is a ı-linea iza ion o i •'is injec i e, •e e y a c _ AiBiis such ha H1._ AiBi/<ı, •_ AiBi 'AiBi. The ı -linea iza ion is said comple e i he union o he a cs _ AiBi is he whole cu e ; hence, 'is piecewise linea . Lemma 2.7 ([13, Co olla y 4.3]).Le R2 be a con ex polygon, and le W@ ! R2 be a pa ame ized Jo dan cu e wi h ini e leng h and le 'W@ !R2 be a ı-linea iza ion o . Then, o e e y P; Q 2@, one has '.@/'.P /; '.Q/ .@/ .P /; .Q/C2ı: In pa icula , o e e y 2Œ0; 2/, we deduce ‰.'/ ‰. / C2ıH1.@/: d. campbell, a. kau anen and e. adici 780 We conclude his sec ion ecalling wo ex ension esul s ha will be use ul in he sequel. The nex p oposi ion (P oposi ion 2.8) is p o ed in [13, Theo em A], and Co olla y 2.10 is a s aigh o wa d consequence o P oposi ion 2.8. P oposi ion 2.8 (Minimal ex ension o s anda d Manha an no m).Le RR2 be a ec angle o he o m Œa; aCŒb; bC , and le 'W@R!R2 be a con inuous injec i e map. Then, o e e y ">0 , he e exis s a piecewise a ine homeomo phism WR!R2coinciding wi h 'on @Rsuch ha kD k0.R/‰0.'/ C": Mo eo e , i ' is piecewise linea , hen he map can be chosen ini ely piecewise a ine. Theo em 2.9 (Minimal ex ension o s anda d Manha an no m).Le ">0 , and le be he mapping om P oposi ion 2.8. Then, jD1 j.Q/ZbC b P'.H1 /; '.H2 /d C"; jD2 j.Q/ZaC a P'.V 1 s/; '.V 2 s/ds C": P oo . The ini ely piecewise a ine homeomo phisms om a ec angle o a polygon in [13] used in he p oo o P oposi ion 2.8 a e cons uc ed in [13, Lemma 2.12]. The key es ima es we need o ex ac a e he las wo unnumbe ed equa ions o he p oo , ound in [13, p. 543]. They say exac ly ha jD1 j.Q/ZbC b P'.H1 /; '.H2 /d C"; jD2 j.Q/ZaC a P'.V 1 s/; '.V 2 s/ds C": Co olla y 2.10 ( W1;1 ex ension wi h non-op imal bound).The e exis s z C > 0 such ha he ollowing holds. Le RR2 be a ec angle, le @R be i s bounda y and le 'W@R!R2 be a con inuous, piecewise linea and injec i e map. Then, he e exis s a ini ely piecewise a ine homeomo phism WR!R2ex ending 'such ha kD kL1.R/z CH1.@R/H1'.@R/: P oo . The conclusion ollows by applying P oposi ion 2.8 wi h "DH1.@R/H1'.@R/ minimal ex ension o he ˛-manha an no m 787 C S0 P PC P0 x1 H2 1 H1 1H1 1 H2 1 V1 s V3 s V2 s V3 s V2 s V1 s Figu e 3. The igu e shows he slicing o he se  in o C[S0 by he ho izon al line R ¹ 1º and Pin o PC[P0by he modi ied geodesic called x1. numbe M , whe e MM1CM2CM3 . We de ine he s ips SiD.RŒ i; iC1/  . They a e all con ex quad ila e als wi h wo ho izon al sides. S ep II. De ini ion o he cu e x1D x'. .R ¹ 1º// and he polygons C[S0D and PC[P0DP .The goal o his s ep is o de ine he piecewise linea cu e x1 , in e nal o P , which will be he image o he segmen H1 1H2 1 in a map x' ex ending ' . The p ecise pa ame iza ion o x'will be p esen ed in he nex s ep; he e we only aim o de ine he cu e x1P. Ou a gumen is ecu si e and so we deal wi h he i s cu e x1 de ined on H1 1H2 1 sepa a ing  in o S0 and  RŒ 1; h DC (see Figu e 3). Simila ly, he cu e x1 means di iding he polygon P in o wo u he polygons: a polygon P0 (which will be he image o S0 ) con aining he cu e '.H1 0H2 0/ and ano he polygon PC (which will be he image o C) (see Figu e 3). Since P is a non-degene a e polygon, le x ı.P/>0 be he pa ame e o De ini ion 2.2 and le ı1> 0 be so small ha ı1<min ²x ı.P/; ". 1 0/ 8hH1.@P/;h 23;" 2³ and Lemma 2.4 applies wi h ı1 o Pand ". 1 0/ 8h.` Ch/: We de ine x1as a ı1-modi ica ion o he geodesic in Pconnec ing H1 1and H2 1: S ep III. De ini ion o x' on @S0 .In his s ep, we ca e abou he de ini ion o x' on @S0 . Mo e p ecisely, we le x'D'on @ and we speci y he pa ame iza ion x'W@C R ¹ 1º! x1 so ha x'is con inuous, injec i e and piecewise linea , and (3.8) ‰0.x'e@S0/C‰0.x'e@C/‰0.'/ C" 2h. 1 0/: d. campbell, a. kau anen and e. adici 788 Le us obse e ha hanks o Lemma 2.7 i is enough o look o a con inuous and injec i e pa ame iza ion W@C[@S0!R2 coinciding wi h ' on @ such ha (3.8) holds o wi h e o " 4h . 1 0/; namely, ‰0. e@S0/C‰0. e@C/<‰0.'/ C" 4h. 1 0/: Indeed, he co ec x' can be ound as a ı -linea iza ion o o some ı small enough depending on " 4h . 1 0/such ha ‰0.x'e@S0/C‰0.x'e@C/<‰0. e@S0/C‰0. e@C/C" 4h. 1 0/: Thanks o ou choice o ı1 and he ac ha x1 is a ı1 -modi ica ion wi h a iable endpoin s o he geodesic connec ing H1 1 and H2 1 , hence spli ing P in o he wo polygons P0and PC, we can apply Lemma 2.4 o ge ha (3.9) P0H1 ;H2 PH1 ;H2 C". 1 0/ 8h.` Ch/ o any 0< < 1; PCH1 ;H2 PH1 ;H2 C". 1 0/ 8h.` Ch/ o any 1< < M: Fo sho , deno e c1D.H1 1/1 and c2D.H2 1/1 . Then, 0c1< c2` . Fo e e y 0 < s < ` , we call s he geodesic inside P connec ing V1 s and V2 s . Mo eo e , whene e c1< s < c2 , we also se V3 sWD .s; 1/ he poin in he in e sec ion o H1 1H2 1 wi h V1 sV2 s . Fo e e y s2.0; c1/[.c2; `/ , we ha e ha ei he V1 s; V 2 s2S0 and using Lemma 2.4, P0.V1 s;V2 s/P.V1 s;V2 s/C". 1 0/ 8h.` Ch/; o V1 s; V 2 s2Cand by Lemma 2.4, PC.V1 s;V2 s/P.V1 s;V2 s/C". 1 0/ 8h.` Ch/: The wo equa ions abo e can be exp essed simul aneously as (3.10) max ®P0.V1 s;V2 s/; PC.V1 s;V2 s/¯P.V1 s;V2 s/C". 1 0/ 8h.` Ch/ o all s2.0; c1/[.c2; `/. On he o he hand, whene e s2.c1;c2/ , he poin s V1 s2P0 and V2 s2PC ; hus, he geodesic s necessa ily in e sec s x1 . Le  be he (injec i e and con inuous) cons an - speed pa ame iza ion o x1 om Œ0; H1.x1/ , .0/ DH1 1 and .H1.x1// DH2 1 . Fo e e y s2.c1; c2/, we hen le X.s/ be he poin in s x1such ha X.s/ Dmax ®x20; H1.x1/W.x/ 2s x1¯: minimal ex ension o he ˛-manha an no m 789 Then, hanks o Lemma 2.3, i is easy o see ha he map s!1.X.s// is non- dec easing; he e o e, i c1< s < s0< c2, hen X.s0/21.X.s//; H1.x1/: No ice ha , in gene al, he unc ion s!1ıX.s/ is no con inuous, no injec i e no su jec i e. Howe e , any one-dimensional mono one unc ion can be app oxima ed uni o mly by s ic ly mono one unc ions. Fu he , i is always possible o sligh ly modi y hese s ic ly mono one app oxima ions in such a way ha hey become con inuous and he p ice o his is loosing he con ol on he uni o m dis ance om he o iginal unc ion on a subse whose measu e can be made as small as desi ed. Then, o e e y  > 0 , i is always possible o ind a con inuous bijec ion X o Œc1; c2 on o x1 such ha (3.11) H1.J/< whe e JWD ®s2.c1; c2/WˇˇX.s/ X.s/ˇˇ> ¯: We can hen ix Dı1 2and apply Lemma 2.4 o ge ha (3.12) P0V1 s;Xı1 2 .s/CPCXı1 2 .s/; V2 sP.V1 s;V2 s/C". 1 0/ 8h.` Ch/ o all s2.c1; c2/nJı1 2 . On he o he hand, we ha e he i ial es ima e (3.13) P0V1 s;Xı1 2 .s/CPCXı1 2 .s/; V2 sH1.@P/CH1.x1/<2H1.@P/ o all s2Jı1 2 . We de ine W@C[@S0!R2 as D' on @ and .V 3 s/DXı1 2 .s/ o e e y s2.c1; c2/ . In pa icula , is con inuous and injec i e and ails o be piecewise linea only on he segmen H1 1H2 1 . Mo eo e , ga he ing oge he (3.9) , (3.10) , (3.12) and (3.13), we deduce ha ‰0. e@S0/C‰0. e@C/ DZ 1 0 P0 .H1 /; .H2 /d CZh 1 PC .H1 /; .H2 /d CZ.0;c1/[.c2;`/ min ®P0.V1 s;V2 s/CPC.V1 s;V2 s/¯ CZ.c1;c2/nJı1 2 P0 .V 1 s/; .V 3 s/CPC .V 3 s/; .V 2 s/ CZJı1 2 P0 .V 2 s/; .V 3 s/CPC .V 3 s/; .V 2 s/ d. campbell, a. kau anen and e. adici 790 Zh 0 P'.H1 /; '.H2 /d CZ.0;`/nJı1 2 P'.V 1 s/; '.V 2 s/ds C". 1 0/ 8h.` Ch/`ChH1Jı1 2CH1Jı1 22H1.@P/ ‰0.'/ C" 4h. 1 0/; whe e in he las inequali y we used (3.11) and he ac ha ı1<". 1 0/ 8hH1.@P/. Finally, hanks o Lemma 2.7 and he conside a ions o he i s pa o he s ep, we can ind a unc ion x'W@C[@S0!R2 ha is con inuous, injec i e and piecewise linea and such ha (3.8) holds. S ep IV. De ini ion o z' on @T 1 0[@R0[@T 2 0 .In his s ep, we u he subdi ide he s ip S0 in he essen ially disjoin union o wo iangles T1 0 , T2 0 and a ec angle R0 wi h he ollowing p ope ies. The ec angle R0 is he bigges ec angle wi h ho izon al and e ical sides inside S0 , such ha he ho izon al sides a e con ained in @S0 , while T1 0,T2 0a e he wo disjoin igh -angle iangles con aining I1 0,I2 0, espec i ely. We con inue o de ine some new z'W@C[@T 1 0[@R0[@T 2 0!R2 coinciding wi h x' on @C[@S0 such ha z' is injec i e, con inuous and piecewise linea and sa is ies he ollowing es ima e: (3.14) ‰0.z'e@T 1 0/C‰0.z'e@R0/C‰0.z'e@T 2 0/‰0.x'e@S0/C" 2h. 1 0/: Le us emphasize ha z'will be de ined so ha z'.@T 1 0[@R0[@T 2 0/P0. We deno e by d1 and d2 he wo alues such ha he p ojec ion o @S0 on o R ¹0º is exac ly Œd1;d2 ¹0º , and we call x1 and x2 hose alues o which he p ojec ion o @R0 on o R ¹0º is he segmen Œx1; x2 ¹0º . No ice ha d1c1x1< x2c2d2 . Since P0 is a non-degene a e polygon, we le x ı.P0/ be he pa ame e o De ini- ion 2.2 and ake ı0 1<min ²x ı.P0/; ". 1 0/ 32hH1.@P0/³ and Lemma 2.4 applies wi h ı0 1 o P0and ". 1 0/ 16h.`Ch/ . Le now x1 be he geodesic inside P0 connec ing x'.V 1 x1/ and x'.V 3 x1/ and le xx1 be i s ı0 1 -modi ica ion in he sense o De ini ion 2.2. In pa icula , xx1 spli s P0 in o wo non-degene a e polygons P1 0 and U , whe e P1 0 con ains x'.I 1 0/ and U con ains x'.I 2 0/ , whe e I1;2 0 a e he wo non-ho izon al segmen s o @S0 @ . This si ua ion is depic ed in Figu e 4. minimal ex ension o he ˛-manha an no m 791 20 d. campbell, a. kau anen and e. adici 𝑇1 0 𝑇2 0 𝑅0 𝐻1 𝑡1 𝐻2 𝑡1 𝐻1 𝑡 𝐻2 𝑡 𝐻3 𝑡 U P1 0 𝑉1 𝑥1 𝑉3 𝑥1 ¯𝜈𝑥1 H1 𝑡 H2 𝑡 Y(𝑡) Figu e 4. The se s 𝑇1 0, 𝑇2 0and 𝑅0, and P1 0and Uin S ep IV. while o he emaining 𝑡∈𝐽𝑙 𝛿′ 1 2 we ge (3.19) 𝜌P1 0𝜓1(𝐻1 𝑡), 𝜓1(𝐻3 𝑡)+𝜌U𝜓1(𝐻3 𝑡), 𝜓1(𝐻2 𝑡),≤ H1(¯𝜈𝑥1) + H1(𝜕P0) ≤2H1(𝜕P0). Then (3.16), (3.17), (3.18) and (3.19) gi e Ψ0(𝜓1 ⌉𝜕𝑇1 0 )+Ψ0(𝜓1 ⌉𝜕(𝑆0 𝑇1 0)) ≤∫𝑐1 𝑑1 𝜌P0¯𝜑(𝑉1 𝑠),¯𝜑(𝑉2 𝑠)+𝜀(𝑡1−𝑡0) 16ℎ(ℓ+ℎ)𝑑𝑠 +∫𝑐2 𝑐1 𝜌P0¯𝜑(𝑉1 𝑠),¯𝜑(𝑉3 𝑠)+𝜀(𝑡1−𝑡0) 16ℎ(ℓ+ℎ)𝑑𝑠 +∫𝑑2 𝑐2 𝜌P0¯𝜑(𝑉1 𝑠),¯𝜑(𝑉2 𝑠)+𝜀(𝑡1−𝑡0) 16ℎ(ℓ+ℎ)𝑑𝑠 +∫(0,𝑡1) 𝐽1 𝛿′ 1 2 𝜌P0¯𝜑(𝐻1 𝑡),¯𝜑(𝐻2 𝑡)+𝜀(𝑡1−𝑡0) 16ℎ(ℓ+ℎ)𝑑𝑡 +2H1𝐽1 𝛿′ 1 2H1(𝜕P0) ≤Ψ0(¯𝜑⌉ P0) + 𝜀 8ℎ(𝑡1−𝑡0), and, as in s ep III, Lemma 2.7 ensu es ha he e is some con inuous, injec i e and piecewise linea map ˜𝜑1:𝜕𝑇1 0∪𝜕𝑆0→R2such ha ˜𝜑1=𝜓1=¯𝜑on 𝜕𝑆0and Ψ0(˜𝜑1 ⌉𝜕𝑇1 0 ) + Ψ0(˜𝜑1 ⌉𝜕(𝑆0 𝑇1 0)) ≤ Ψ0(¯𝜑⌉ P0) + 𝜀 4ℎ(𝑡1−𝑡0). To conclude he s ep we need o epea he e y same a gumen on ˜𝜑1 ⌉𝜕(𝑆0 𝑇1 0) by eplacing P0 wi h U and conside ing a 𝛿′′ 1 -modi ica ion o 𝜈𝑥2 , whe e 𝛿′′ 1 is chosen so Figu e 4. The se s T1 0,T2 0,R0,P1 0and Uin S ep IV. Thanks o Lemma 2.4, we ha e (3.15) Ux'.V 1 s/; x'.V 3 s/P0x'.V 1 s/; x'.V 3 s/C". 1 0/ 16h.`Ch/; o all s2.x1; x2/; Ux'.V 1 s/; x'.V 2 s/P0x'.V 1 s/; x'.V 2 s/C". 1 0/ 16h.`Ch/; o all s2.x2; d2/; while o all s2.d1; x1/, (3.16) P1 0x'.V 1 s/; x'.V 2 s/P0x'.V 1 s/; x'.V 2 s/C". 1 0/ 16h.` Ch/: We con inue simila ly as desc ibed in S ep III. Fo e e y 2.0; 1/ , we deno e he poin x'.H1 /DH1 2P1 0 and x'.H2 /DH2 2U ; hus, he geodesic  connec ing H1 and H2 inside P0 mus in e sec xx1 . So also in his case, o e e y 2.0; 1/ , we can ind a map Y. / iden i ying he las poin o he in e sec ion  xx1 unning xx1 om x'.V 1 x1/ o x'.V 3 x1/ . Mo eo e , exac ly as explained in S ep III, we can ind a con inuous and injec i e app oxima ion Yı0 1 2 such ha ˇˇY. / Yı0 1 2 . /ˇˇ<ı0 1 2 o all 2.0; 1/nJ1 ı0 1 2 and H1J1 ı0 1 2<ı0 1 2: So i we now call H3 WD . ; x1/ , hen we can de ine 1W@T 1 0[@S0!R2 in his way: 1D x'on @S0and 1.H3 /DYı0 1 2 . / o all 2.0; 1/: Then, he map 1 is con inuous and injec i e and ails o be piecewise linea only on he segmen ¹x1º  Œ0; 1 . Using Lemma 2.4, o all 2.0; 1/nJ1 ı0 1 2 , we can es ima e P1 0 1.H1 /; 1.H3 /CU 1.H3 /; 1.H2 / P0x'.H1 /; x'.H2 /C". 1 0/ 16h.` Ch/; (3.17) d. campbell, a. kau anen and e. adici 792 while o he emaining 2Jl ı0 1 2 , we ge P1 0 1.H1 /; 1.H3 /CU 1.H3 /; 1.H2 / H1.xx1/CH1.@P0/2H1.@P0/: (3.18) Then, (3.15), (3.16), (3.17) and (3.18) gi e ‰0 1 e@T 1 0C‰0 1 [email protected] 0/ Zc1 d1 P0x'.V 1 s/; x'.V 2 s/C". 1 0/ 16h.` Ch/ds CZc2 c1 P0x'.V 1 s/; x'.V 3 s/C". 1 0/ 16h.` Ch/ds CZd2 c2 P0x'.V 1 s/; x'.V 2 s/C". 1 0/ 16h.` Ch/ds CZ.0; 1/nJ1 ı0 1 2 P0x'.H1 /; x'.H2 /C". 1 0/ 16h.` Ch/d C2H1J1 ı0 1 2H1.@P0/ ‰0.x'eP0/C" 8h. 1 0/; and, as in S ep III, Lemma 2.7 ensu es ha he e is some con inuous, injec i e and piecewise linea map z'1W@T 1 0[@S0!R2such ha z'1D 1D x'on @S0and ‰0z'1 e@T 1 0C‰0z'1 [email protected] 0/‰0.x'eP0/C" 4h. 1 0/: To conclude he s ep, we need o epea he e y same a gumen on z'1 [email protected] 0/ by eplacing P0 wi h U and conside ing a ı00 1 -modi ica ion o x2 , whe e ı00 1 is chosen so ha ı00 1<min ²x ı.U/; ". 1 0/ 32`H1.@U/³ and Lemma 2.4 applies wi h ı00 1 o Uand ". 1 0/ 16h.`Ch/ . This would p o ide a con inuous, injec i e and piecewise linea map z'W@T 1 0[ @R0[@T 2 0!R2ex ending z'1(hence, ul ima ely, x') such ha ‰0.z'e@T 1 0/C‰0.z'e@R0/C‰0.z'e@T 2 0/‰0z'1 [email protected] 0/C" 4h. 1 0/ ‰0.x'eP0/C" 2h. 1 0/ hus p o ing (3.14) and concluding he s ep. minimal ex ension o he ˛-manha an no m 793 S ep V. Recu sion and conclusion. In his inal s ep, we wan o conclude ou cons uc ion by ecu sion. In S eps III and IV, we di ided  in o a new con ex polygon wi h wo ho izon al sides C and a ho izon al s ip S0 gi en by a ec angle R0 and wo iangles T1 0,T2 0. Then, we de ined con inuous, injec i e and piecewise linea unc ions x'; z' such ha x'D z'on @Csa is ies (by (3.8), (3.14) and ou choice o ı1) ‰0.x'e@S0/C‰0.x'e@C/‰0.'/ C" 2h. 1 0C2ı1/ ‰0.'/ C" 2h. 1 0/C" 4 1 2; ‰0.z'e@T 1 0/C‰0.z'e@R0/C‰0.z'e@T 2 0/‰0.x'e@S0/C" 2h. 1 0C2ı1/ ‰0.x'e@S0/C" 2h. 1 0/C" 4 1 2: I e a ing he cons uc ion and choosing o e e y i he pa ame e ıi sui ably small depending on i1,h 2iC1and he polygon PCDPnSi1 jD0Pj, we hen ind M1 X iD0 ‰0.x'e@Si/‰0.'/ C" 2h M1 X iD0 . iC1 i/C" 4 M1 X iD0 1 2i and M1 X iD0 ‰0.z'e@T 1 i/C‰0.z'e@Ri/C‰0.z'e@T 2 i/  M1 X iD0 ‰0.x'e@Si/C" 2h M1 X iD0 . iC1 i/C" 4 M1 X iD0 1 2i; which inally imply (3.6) and (3.7), espec i ely. 4. Piecewise a ine ex ension In his sec ion, we in es iga e wo possible ini ely piecewise a ine homeomo phic ex ension inside iangles. Lemma 4.1 (Ex ension-di ec ).Le TR2 be a iangle o co ne s A; B; C such ha BC is ho izon al and A is he in e sec ion o BC wi h he bisec o o he angle a A . I 'W@T !R2 is con inuous, injec i e and linea on each o he segmen s AB; AC; BA and AC , hen he e exis s a bi-a ine homeomo phism WT!R2 such ha D' on @T and kD k0.T / H1'.@T /H1.@T /: d. campbell, a. kau anen and e. adici 794 P oo . The p oo is immedia e, and indeed i is enough o conside he con inuous map which is a ine on each o he iangles T1WD ABA,T2WD BAC. Then, we can compu e D1 eT1D'.A/'.B/ .A/1.B/1 L2; D1 eT2D'.C / '.A/ .C /1.A/1 L2 and D2 eT1D'.A/ '.A/D1 eT1.A/1.A/1 .A/2.A/2 L2; D2 eT2D'.A/ '.A/D1 eT2.A/1.A/1 .A/2.A/2 L2: In pa icula , one can es ima e kD k0.T / D jD1 j.T1/C jD2 j.T1/C jD1 j.T2/C jD2 j.T2/ 1 2ˇˇ'.A/'.B/ˇˇˇˇ.A/2.A/2ˇˇCˇˇ.A/1.A/1ˇˇ C1 2ˇˇ'.A/'.A/ˇˇˇˇ.B/1.A/1ˇˇ C1 2ˇˇ'.C / '.A/ˇˇˇˇ.A/2.A/2ˇˇCˇˇ.A/1.A/1ˇˇ C1 2ˇˇ'.A/'.A/ˇˇˇˇ.C /1.A/1ˇˇ H1'.@T /1 24jAAjCjBCj H1'.@T /1 22jABj C 2jACj C 2jBCj H1'.@T /H1.@T /: Lemma 4.2 (Ex ension-indi ec ).Le TR2 be a iangle o co ne s A; B; C such ha AB is ho izon al, BC is e ical and le 'W@T !R2 be a con inuous, piecewise linea , injec i e map such ha ' is linea on he hypo enuse AC . Fo e e y ">0 , he e exis s a ini ely piecewise a ine homeomo phism WT!R2 such ha D' on @T and kD k0.T / ‰0.'/ C242H1.@T /H1'.AC /C": P oo . Le ">0be ixed a bi a y small. Fo simplici y o no a ion, h ough he p oo we e e o ˇ as he in e nal angle o he co ne AD.0; 0/ and we will deno e dWD j'.A/ '.C /j D H1.'.AC // and dD jACj. Clea ly, since he in e nal angle in Bis =2, hen ˇ2.0; =2/. minimal ex ension o he ˛-manha an no m 795 X Y Z A B C '.A/ '.C / D E F z T G H1 H2 H3 V3 s V2 s V1 s PADEC P Pz T Q  z'.D/'.E/ P '.D/ '.F / ˇ Figu e 5. The decomposi ion o T in o TD[z T[ADEC , and he co esponding decom- posi ion o in '.@T / in o P[Pz T[PADEC . Since he polygon o bounda y '.@T / is non-degene a e, hen De ini ion 2.2 p o ides some cons an x ı > 0, and hen we conside (4.1)  < ²1; d 2; x ı 4;d 14;" 4;1C1 2 an ˇ1 ;kD'k1 1³: The basic idea o he p oo is o ind a sui able one-dimensional skele on ‡ inside T , cons uc a con inuous, piecewise linea and injec i e map z'W‡!R2 coinciding wi h ' on @T and inally pe o m a sui able piecewise a ine ex ension inside each componen o he pa i ion o Tiden i ied by ‡. Fo cla i y, we p esen he p oo in h ee sepa a e s eps. S ep I. De ini ion o a i s skele on „ and a con inuous piecewise linea injec i e map '1W„!R2 .In his s ep, we would like o cons uc a one-dimensional skele on o he o m „D@T [DE [F G , o some sui ably chosen poin s D; E; F; G , and we will de ine an ex ension '1 o ' on DE [F G ha is s ill con inuous, piecewise linea and injec i e. See Figu e 5 o an illus a ion. We will selec D; E; F; G so ha D2AB; E; F 2BC; G 2DE; F G kAB and DE kAC; sa is ying he ollowing es ima es: (4.2) jADj<  and jCEj< ; H1'1.@ z T /< 4; ‰0.'1 e@/‰0.'/ C.dC14/H1.@T /; whe e z TT is he iangle o co ne s E; F; G and T is he apezoid o co ne s D; B; F; G. d. campbell, a. kau anen and e. adici 796 Ha ing ixed  , by he assump ions on ' , we can choose D2AB and E2BC so ha (i) jADj<  and jCEj< ; (ii) j'.A/ '.D/j<  and j'.C / '.E/j< ; (iii) he es ic ion o 'is linea on AD and CE; (i ) DE is pa allel o AC ; ( ) he poin X is on he in e nal bisec o o '.A/ and Y on he in e nal bisec o o '.C / such ha j'.A/ Xj< 2 and j'.C / Yj< 2 and he piecewise linea pa h z'.D/'.E/ WD '.D/XY'.E/ lies in he in e io o '.@T / and is a x ı=2-modi ica ion o he geodesic '.D/'.E/ in he sense o De ini ion 2.2. Obse e ha (ii) and ( ) imply ha j'.D/ Xj;j'.E/ Yj< 3 and also H1.z'.D/'.E// < j'.D/ XjCjXYjCj'.E/ Yj < 6 C j'.A/ XjCj'.A/ '.C /jCj'.C / Yj <dC10: (4.3) We ind a poin F on he segmen EB , a poin G2DE and a poin Z2ŒY'.E/ such ha ( i) jFEj<  and j'.F / '.E/j< ; ( ii) 'is linea on EF ; ( iii) j'.E/ Zj<  and Œ'.F /Zlies in in '.@T /, and as a consequence, j'.F / Zj< 2; (ix) G2DF2,jGEj< .sin ˇ/1and jGFj< . an ˇ/1. This concludes he de ini ion o „ . Indeed, (i) ensu es he i s equa ion o (4.2) , hen z Tis a igh -angle iangle and is a apezoid inside T. We now p oceed o cons uc a unc ion '1W„!R2 ex ending ' such ha he second and hi d es ima es o (4.2) a e sa is ied. In o de o do ha , we conside wo u he auxilia y poin s P; Q 2DG so ha (x) jPDj<  and jQGj< ; and we se '1.P / WD X; '1.Q/ WD Yand '1.G/ WD Z: We hen de ine '1W„!R2 so ha '1D' on @T , '1 eDP is he pa ame iza ion a cons an speed o he segmen '.D/X , '1 ePQ is he pa ame iza ion a cons an speed o he segmen XY , '1 eQG is he pa ame iza ion a cons an speed o he segmen YZ , '1 eGE is he pa ame iza ion a cons an speed o he segmen Z'.E/ and, inally, '1 eGF is he pa ame iza ion a cons an speed o he segmen Z'.F / . A ske ch o he si ua ion is p esen ed in Figu e 5. minimal ex ension o he ˛-manha an no m 803 We obse e ha is con inuous and injec i e, hence a homeomo phism, since D z' on ‡ . F om he same obse a ion, we also deduce ha D' on @T because z'D' he e. Ga he ing oge he (4.10), (4.11), (4.12) and (4.7), we ind kD k0.T / D kD!k0.P/C kDwk0.z T / C M1 X iD0 kD ik0.Ri/C kD k0.y T / z C ddC21C1 2 an ˇC M1 X iD0‰0.z'e@Ri/C" 2iCd" z C ddC21C1 2 an ˇC‰0.'/ CCdH1.@T / Cz C " ‰0.'/ Cz CH1.@T /H1'.AC /Cz C "; whe e in he las inequali y, we used ha dD jACj  H1.@T / and dDˇˇ'A'.C /ˇˇ: 5. P oo o Theo ems 1.1 and 1.2 P oo o Theo em 1.1.Le ">0be a bi a y ixed and le (5.1) 0 <  < min ²1; " z CCz CH1.Q/³ o some la ge app op ia e bu ixed geome ic cons an z C . We desc ibe in de ail he p oo when Q is o class (iii) in he sense o Rema k 3.1, while he o he cases a e an ob ious modi ica ion o he cu en a gumen . Applying Lemma 3.2 o Q; '; ˛ and he pa ame e  , we can pa i ion Q in wo iangles T1; T2 and a con ex polygon  and ind a con inuous, piecewise linea , injec i e map x'W@T1[@T2[@ !R2 wi h he p ope ies lis ed in he s a emen o Lemma 3.2. In pa icula , om (3.3), i ollows ha (5.2) ‰˛.x'e@/‰˛.'/ C; whe e (3.1) and (3.2) ensu e ha (5.3) H1.@T1/CH1.@T2/ < ; H1x'.@T1/CH1x'.@T2/< : Fu he mo e, since  is a con ex polygon wi h wo pa allel sides in di ec ion ˛ , we can apply he ˛ - o a ed e sion o Lemma 3.3 o  , x'1 and he pa ame e  o ind M inc easing alues . i/M1 iD0 and ˛ - o a ed s ips Si , which can be seen as he union o a d. campbell, a. kau anen and e. adici 804 ec angle Ri and wo iangles T1 i and T2 i , and a con inuous piecewise linea injec i e map O'WSM1 iD0@T 1 i[@Ri[@T 2 i!R2 coinciding wi h x' on @ wi h he p ope ies o Lemma 3.3. In pa icula , hanks o (3.7), we deduce (5.4) M1 X iD0‰˛O'e@T 1 iC‰˛O'e@RiC‰˛O'e@T 2 i‰˛x'1 e@C; while (3.5) ensu es ha (5.5) H1O'.I 1 i/CH1O'.I 2 i/<  whe e I1 iD@T 1 i @ and I2 iD@T 2 i @. We a e inally in posi ion o de ine a unc ion z'W@T1[ M1 [ iD0 @T 1 i[@Ri[@T 2 i![@T2!R2 ha is con inuous, injec i e, ini ely piecewise linea and such ha z'D x'1 on @T1[@T2 and z'D O'on SM1 iD0@T 1 i[@Ri[@T 2 i. Once he e we will pe o m he homeomo phic piecewise a ine ex ension on T1; T2; T 1 i; T 2 i and Ri independen ly o e e y iD0; : : : ; M 1 . We i s ocus on he ex ension inside he s ips Si . Le i2 ¹0;:::;M 1º be ixed; we hen apply he ˛ - o a ed e sion o P oposi ion 2.8 o Ri;z'e@Ri and pa ame e . iC1 i/ o ind ini ely piecewise a ine homeomo phisms iWRi!R2 coinciding wi h z' on @Ri such ha (5.6) kD ik˛.Ri/‰˛.z'e@Ri/C. iC1 i/: By cons uc ion, we ha e ha T1 i; T 2 i a e igh -angle iangles whose hypo enuse is con ained in @ @Q , and we can apply he ˛ - o a ed e sion o Lemma 4.2 o T1;2 i , z'e@T 1;2 i and pa ame e . iC1 i/ o ind ini ely piecewise a ine homeomo phisms w1;2 iWT1;2 i!R2 coinciding wi h z'on @T 1;2 isuch ha kDw1;2 ik˛.T 1;2 i/‰˛z'e@T 1;2 iCz CH1.@T 1;2 i/H1z'.I1;2 i/Cz C . iC1 i/; which hanks o (5.5) implies kDw1 ik˛.T 1 i/C kDw2 ik˛.T 2 i/ ‰˛z'e@T 1 iC‰˛z'e@T 2 iCz CH1.@T 1 i/CH1.@T 2 i/Cz C . iC1 i/: (5.7) minimal ex ension o he ˛-manha an no m 805 Le us also no ice ha , o u u e need, since he iangles T1;2 i ha e one angle equal o =2 and, by cons uc ion, hei hypo enuse is I1;2 i , hen one has H1.@T 1;2 i/ 3H1.I1;2 i/ o e e y i . Mo eo e , ha ing I1;2 i.@ @Q/ and he iangles pai wise essen ially disjoin , we deduce (5.8) M1 X iD0H1.@T 1 i/CH1.@T 2 i/3 M1 X iD0H1.I1 i/CH1.I2 i/3H1.@Q/: Le us now conside he ex ension inside he iangles T1; T2 . In his case, by cons uc ion, we a e in posi ion o apply he ˛ - o a ed e sion o Lemma 4.1 o T1;2 and z'e@T1;2 o ind bi-a ine homeomo phisms w1;2 WT1;2 !R2such ha kDw1;2k˛.T1;2/H1z'.@T1;2/H1@T1;2 which, hanks o (5.3), gi es (5.9) kDw1k˛.T1/C kDw2k˛.T2/ < 22: We can inally de ine WQ!R2 o be he piecewise a ine unc ion such ha Dw1;2 on T1;2; Dw1;2 ion T1;2 iand D ion Ri o e e y iD0;: : :;M 1: By cons uc ion, is con inuous because D z' on he one-dimensional skele on @T1[.SM1 iD0@T 1 i[@Ri[@T 2 i/[@T2 , and, mo eo e , coincides wi h z'D' on @Q . To conclude, i is only le o e i y he alidi y o (1.4) , bu his is now a s aigh o wa d consequence o (5.6) , (5.7) , (5.9) , (5.4) , (5.8) and (5.2) . Indeed, we ge kD k˛.Q/D kDw1k˛.T1/C kDw2k˛.T2/ C M1 X iD0kDw1 ik˛.T 1 i/C kD ik˛.Ri/C kDw2 ik˛.T 2 i/  M1 X iD0‰˛O'e@T 1 iC‰˛O'e@RiC‰˛O'e@T 2 i C42Cz C  M1 X iD0H1.@T 1 i/CH1.@T 2 i/C. iC1 i/ ‰˛.x'1 e@/CC42Cz C H1.@Q/Cdiam Q ‰˛.x'e@/C6 Cz C H1.@Q/; ‰˛.'/ Cz C 1CH1.@Q/; and hen es ima e (1.4) ollows since has been chosen as in (5.1). d. campbell, a. kau anen and e. adici 806 5.1. P oo o Theo em 1.2 To p o e he claim, i su ices o epea he p oo o he abo e lemmas as be o e bu using he es ima es om Theo em 2.9 ins ead o om P oposi ion 2.8. In all o ou calcula ions, we es ima e RP.'1.H2 /;'1.H 3 //d and RB1 D1P.'1.V 1 s/;'1.V 2 s//ds sepa a ely. Now i su ices o keep hem sepa a e ins ead o summing hem. The key es ima es in Lemma 3.2 a e (3.4) , in Lemma 3.3 hey a e (3.12) , (3.13) and he calcula ion ollowing, and in Lemma 4.2 hey a e (4.4), (4.5), (4.6). 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Zbl 1359.46036 MR 3519110 Recei ed 20 Janua y 2023, and in e ised o m 30 Augus 2023 Daniel Campbell Depa men o Ma hema ical Analysis, Cha les Uni e si y Sokolo ská 83, 186 00 P ague 8, Czech Republic [email p o ec ed] Aapo Kau anen Depa men o Ma hema ics and S a is ics, Uni e si y o Jy äskylä PL 35, 40014 Jy äsklyän yliopis o, Finland [email p o ec ed] Emanuela Radici Depa men o In o ma ion Enginee ing, Compu e Science and Ma hema ics (DISIM), Uni e si y o L’Aquila Via Ve oio 1 (Coppi o), 67100 L’Aquila, I aly emanuela. adici@uni aq.i