Minimal extension for the α-Manhattan norm

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Minimal extension for the α-Manhattan norm © 2024 Accademia Nazionale dei Lincei Published version Campbell, Daniel; Kauranen, Aapo; Radici, Emanuela Campbell, D., Kauranen, A., & Radici, E. (2023). Minimal extension for the α-Manhattan norm. Rendiconti Lincei: Matematica e Applicazioni, 34(4), 773-807. https://doi.org/10.4171/rlm/1027 2023 Rend. Lincei Mat. Appl. 34 (2023), 773–807 DOI 10.4171/RLM/1027 ©2024 Accademia Nazionale dei Lincei Published by EMS Press This work licensed under a CC BY 4.0 license Calculus of Variations. – Minimal extension for the ˛ -Manhattan norm, by Daniel Campbell, Aapo Kauranen and Emanuela Radici, communicated on 10 November 2023. Abstract. – Let @Q be the boundary of a convex polygon in R2 , e˛D.cos ˛; sin ˛/ and e? ˛D.sin ˛; cos ˛/ a basis of R2 for some ˛2Œ0; 2/ and 'W@Q!R2 a continuous, finitely piecewise linear injective map. We construct a finitely piecewise affine homeomorphism vWQ!R2 coinciding with ' on @Q such that the following property holds: jhDv; e˛ij.Q/ (resp., hDv; e? ˛ij.Q/ ) is as close as we want to inf jhDu; e˛ij.Q/ (resp., inf jhDu; e? ˛ij.Q/ ) where the infimum is meant over the class of all BV homeomorphisms u extending ' inside Q . This result extends that already proven by Pratelli and the third author in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 3, 511–555] in the shape of the domain. Keywords. – Homeomorphic extension, BV homeomorphisms, strict approximation in BV. Mathematics Subject Classification 2020. – 46E35 (primary); 30E10, 58E20 (secondary). 1. Introduction In this paper, we are interested in the problem of extending injective continuous and piecewise linear boundary values from a convex polygon by piecewise affine homeomorphisms. The motivation for such a study arises in the context of approximation problems found in regularity theory for non-linear elasticity. There is already a plurality of extension results in a variety of contexts, which have been applied to solve various approximation problems. Let us now give an overview of some examples. In general, we are interested in the approximation of a weakly differentiable homeomorphism, which we would like to approximate by C1 homeomorphisms or by locally finite piecewise affine homeomorphisms. The approximation of a planar W1;p homeomorphism 1 < p < 1 in [9,10] relies heavily on the injectivity of the harmonic extension of convex boundary values. In [5], the authors were also able to approximate a bi-Lipschitz map and its inverse simultaneously in the .p; p/ -bi-Sobolev setting and to do so used the extension result in [6]. In order to solve the W1;1 case in [8], the authors had to develop an independent extension result in that paper which was further examined and improved in [1,15]. The extension result was also utilized in d. campbell, a. kauranen and e. radici 774 the .1; 1/ -bi-Sobolev setting in [12]. Finally, let us mention that the authors of [14] approximate planar BV homeomorphisms using an extension result they proved in [13]. More than just the approximation of weakly differentiable homeomorphisms by diffeomorphisms, these extension results have been key in examining the behavior of weak and strong limits of homeomorphisms in their respective classes. Such results include a categorization of the closure of Hom \W1;p , p2 , in [11], a categorization of the closure of Hom \W1;p,1<p<2, in [7] and partial BV result in [2,4]. It was demonstrated in [13] that their main extension result can be “rotated” to approximate a BV homeomorphism strictly and similarly in [14] for the area-strict case. Nevertheless, this approach makes the application of the extension result somewhat cumbersome and technical. The main result of the present paper is a piecewise affine homeomorphic extension that improves on that of [13]. More precisely, we consider extensions of piecewise linear boundary values defined on a boundary of convex quadrilaterals (and not only rectangles parallel to the coordinate axes as in [13]) which are optimal in a particular BV sense. We emphasize that the generality of the class of convex quadrilaterals includes the “rotated” version of the extension result of [13]. Also, our Theorem 1.2 is stronger than the extension theorem there (not only because of the shape of Q ) in the sense that it immediately implies their extension theorem but the opposite is not true (see Remark 1.3). Nevertheless, this improvement is a case of separating estimates already conducted in [13]. The motivation for our extension theorem is the full categorization result in [3], where we identify a condition which guarantees that a map is a strict or area-strict limit of BV homeomorphisms. In the course of the approximation, we want to work on grids that are not only made up of rectangles, and we prefer to not have to rotate the rectangles. In that sense, we need the current result, which we present below, after we set some necessary notation. Let QR2 be a convex polygon, let ˛2Œ0;2/ be fixed and call e˛D.cos˛;sin˛/ and e? ˛D.sin ˛; cos ˛/. We define the following numbers (see Figure 1): (1.1) aWD inf ®hx; e? ˛i W x2Q¯aCWD sup ®hx; e? ˛i W x2Q¯; bWD inf ®hx; e˛i W x2Q¯; bCWD sup ®hx; e˛i W x2Q¯: For each s2.a; aC/, we define V1 s; V 2 suniquely by the conditions (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: Similarly, for every t2.b; bC/, we define H1 t; H2 tuniquely by (1.3) H1 t; H2 t2@Q;hH1 t; e˛iDhH2 t; e˛i D t; hH1 t; e? ˛i<hH2 t; e? ˛i: minimal extension for the ˛-manhattan norm 775 minimal extension for the 𝛼-manhattan norm 3 𝑒𝛼 {⟨𝑥, 𝑒𝛼⟩=𝑏+} {⟨𝑥, 𝑒𝛼⟩=𝑎−} {⟨𝑥, 𝑒𝛼⟩=𝑏−} {⟨𝑥, 𝑒𝛼⟩=𝑎+} 𝐻1 𝑡 𝑉1 𝑠 𝐻2 𝑡 𝑉2 𝑠 Q Figure 1. Polygon Qwith 𝐻1 𝑡, 𝐻2 𝑡, 𝑉1 𝑠and 𝑉2 𝑠. by 𝜌P(A,B) the geodesic distance between A and B inside P . We define the quantity Ψ𝛼(𝜑):=∫𝑎+ 𝑎− 𝜌P(𝜑(𝑉1 𝑠), 𝜑(𝑉2 𝑠))𝑑𝑠 +∫𝑏+ 𝑏− 𝜌P(𝜑(𝐻1 𝑡), 𝜑(𝐻2 𝑡))𝑑𝑡. Loosely speaking, the quantity Ψ𝛼(𝜑) accounts for the length of all the geodesics inside P connecting pairs of points on 𝜑(𝜕Q) whose preimage in 𝜑 is a pair of points in 𝜕Q lying on a line parallel to either 𝛼 or 𝛼⊥ . Further for 𝑢∈𝐵𝑉 (Ω,R2) we denote the 𝛼-Manhattan norm of 𝐷𝑢 as ∥·∥𝛼which we define as ∥𝐷𝑢∥𝛼(Q) :=|⟨𝐷𝑢, 𝑒𝛼⟩|(Q) + |⟨𝐷𝑢, 𝑒⊥ 𝛼⟩|(Q). The main results of the paper is are the follwoing. Theorem 1.1.Let 𝛼∈ [0,2𝜋) be fixed, Q ⊂ R2 be a convex polygon and 𝜑:𝜕Q → R2 be a continuous piecewise linear injective map. Then for every 𝜀 > 0 there exists a finitely piecewise affine homeomorphism 𝑣:Q → R2extending 𝜑, such that (1.4) ∥𝐷𝑣∥𝛼(Q) ≤ Ψ𝛼(𝜑) + 𝜀. Figure 1. Polygon Qwith H1 t,H2 t,V1 sand V2 s. Let 'W@Q!R2 be continuous, injective and piecewise linear. We denote P as the bounded component of R2n'.@Q/ . For every pair of points A;B2x P , we denote by P.A;B/ the geodesic distance between A and B inside x P . We define the quantity ‰˛.'/ WD ZaC a P'.V 1 s/; '.V 2 s/ds CZbC b P'.H1 t/; '.H2 t/dt: Loosely speaking, the quantity ‰˛.'/ accounts for the length of all the geodesics inside P connecting pairs of points on '.@Q/ whose preimage in ' is a pair of points in @Q lying on a line parallel to either ˛ or ˛? . Further, for u2BV.; R2/ , we denote the ˛-Manhattan norm of Du as kk˛which we define as kDuk˛.Q/WD ˇˇhDu; e˛iˇˇ.Q/CˇˇhDu; e? ˛iˇˇ.Q/: The main results of the paper are the following. Theorem 1.1.Let ˛2Œ0; 2/ be fixed, QR2 a convex polygon and 'W@Q!R2 a continuous piecewise linear injective map. Then, for every ">0 , there exists a finitely piecewise affine homeomorphism vWQ!R2extending ', such that (1.4) kDvk˛.Q/‰˛.'/ C": Theorem 1.2.Let ">0and let vbe the extension from Theorem 1.1. Then, (1.5) ˇˇhDv; e˛iˇˇ.Q/ZbC b P'.H1 t/; '.H2 t/dt C"; ˇˇhDv; e? ˛iˇˇ.Q/ZaC a P'.V 1 s/; '.V 2 s/ds C": d. campbell, a. kauranen and e. radici 776 Remark 1.3.We observe that Theorem 1.1 is stronger than the result of [13] as it provides the almost optimal extension with respect to any rotated Manhattan norm and not just for the canonical one (where ˛D0). Let us also remark that Theorem 1.2 immediately implies Theorem 1.1 but the argument used to construct vis exactly the same. Also, it is immediate that ZbC b P'.H1 t/; '.H2 t/dt inf ®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 inf ®ˇˇhDu; e? ˛iˇˇ.x Q/Wu2Hom \BV.x Q;R2/; u D'on @Q¯; and our result in fact shows that there is a sequence of homeomorphisms achieving the infimum and having variation converging to the left-hand side in the sense of (1.5) . This fact is actually a direct consequence of the proofs in [13], though it was not explicitly remarked there. The key argument is in Theorem 2.9. 1.1. Sketch of the proof Before expounding the proof in detail, let us look at an overview of the proof. We start with a convex polygon Q . Up to a rotation of ˛ , we may assume that ˛D0 . Either (the rotated) Q has horizontal sides, or after removing a tiny triangle called T1 close to the lowest point of Q and a triangle called T2 close to the highest point of Q we get a convex  that has a pair of horizontal sides (see Figure 2). We extend ' on @T1; @T2 so that it is continuous injective and piecewise linear. By making the triangles small enough, we guarantee that ‰0.'.// C‰0.'.T1// C‰0.'.T2// ‰0.'.Q// C" . Here, our new 'extends the original 'from @Q. This step is Lemma 3.2. Now, we separate  into thin horizontal strips Si (see Figure 3), defining a continuous injective piecewise linear ' on @Si so that PM iD1‰0.'.@Si// ‰0.'.@// C" which extends the original 'from @ [@T1[@T2. This step is Lemma 3.3. We separate each Si into a central rectangle and a pair of right-angle triangles at each end. On the rectangular domains Ri , we can use Proposition 2.8 to extend the boundary values and get a piecewise affine homeomorphism wi on the Ri satisfying an estimate on jDwij.Ri/ . In Lemma 4.2, we show how we extend the boundary values to get a piecewise affine homeomorphism on the triangles at the ends of the strips; see Figure 6. We do this by further separating them into even thinner rectangles where minimal extension for the ˛-manhattan norm 777 we can extend and estimate as above. The remaining part of the set is so small that its contribution to the norm is bounded by 2i". The final part of the proof is collating the estimates and summing to estimate that our mapping vsatisfies (1.4). 2. Preliminaries In this section, we recall a list of definitions and known geometrical results which are already available in the literature. Most of them are taken from [13,14]. Notation 2.1.Throughout the paper, we endeavor to keep to the following norms of notation: •Qis a convex polygon, •P is a 2-dimensional polygon with boundary @P . If 'W@Q!R2 is injective and piecewise linear continuous, then P is the polygon corresponding to the bounded component of R2n'.@Q/and PD'.@Q/, •˛2Œ0; 2/ is a given angle and the vector e˛WD .cos ˛; sin ˛/ . Also, we denote e? ˛WD .cos.˛ C=2/; sin.˛ C=2//, •uand vare planar BV mappings, •a;aC; b; bC are the numbers from (1.1) , typically s2.a;aC/ and t2.b; bC/ and `DaCa,hDbCb, •is a convex polygon with 2 sides parallel to ˛, •T; T1; T2;z T ; T 1 i; T 2 iare all triangles, •V1 s; V 2 s; H1 t; H2 t are the points satisfying the conditions in (1.2) and (1.3) although we may replace Qwith another convex polygon, for example, or T, • by RQDŒa; aCe˛CŒb; bCe? ˛ we denote the smallest rectangle with sides parallel to e˛and e? ˛containing Q, •c1; c2; d1; d22Rare ordinates, • by z C we denote a generic constant whose precise value may vary between estimates, •the points in the preimage are A; B; C; D; E; F; G; P; Q,1 • if ' is a piecewise linear injective map defined on the triangle ABC , we call dWD j'.A/ '.C /jthe length of the image of the hypotenuse through ', (1) We do not need to utilize the notation B.x; r/ D ¹yW jyxj< rº so there is no danger of confusion when using Bto denote a point. d. campbell, a. kauranen and e. radici 778 •ˇ2.0;  2/is the angle at Ain the triangle ABC , •>0is a small chosen parameter, •the points in the image are written in bold font, e.g., A;B;C;D;X;Y;Z, •P;P0;PC are polygons in the image, typically the piecewise affine image of a polygon in the preimage, e.g., PD'.@Q/, •' , are continuous injective piecewise linear maps from one-dimensional “skeletons” (i.e., a finite union of segments) in the preimage, • given a set AR2 and a function 'WA!R2 , we denote by 'eB the restriction of 'to a subset BA, •A;B is the geodesic curve from A to B in P and P.A;B/ is the length of that curve. Definition 2.2 (Geodesics and modified geodesics).Let PR2 be a polygon, and let A and B be any two distinct points in P . We define AB as the unique geodesic (i.e., curve of minimal length) connecting them, lying inside P . Notice that AB is a piecewise linear curve, whose vertices are only A;B and some vertices of @P whose internal angles have size at least  . Assume now that A;B2@P , and let W1;W2; : : : ; WK be all the vertices of Pmet by AB, so that AB DAW1;W2;:::;WKB: Fix now any ı > 0 . For every 1iK , let z Wi¤Wi be some arbitrary point in the internal bisector of the angle at Wi having distance from Wi smaller than ı . The piecewise linear curve zAB DAz W1;z W2;:::; z WKBis then called a ı-modification of AB. Notice that there exists a constant x ı.P/>0 , depending on P but not on A and B , such that the interior of zAB is contained in the interior of P if ı < x ı.P/ , unless the segment AB is already contained in @P , in which case KD0 and zAB DAB @P . Lemma 2.3 ([13, Lemma 2.4]).Let A;B;C and D be four distinct points in a polygon P . Then, the intersection AB \CD is either empty or closed and connected. Assume now also that A;B;C;D2@P and call @P1; @P2 the two components of @Pn ¹C;Dº . If A2@P1 and B2@P2 , then AB \CD ¤ ; . If A;B2@P1 and AB \CD ¤ ; , then the first and last point of this intersection must either be vertices of P or coincide with one of the points Aor B. We remark that in the reference the lemma is stated without closedness of the intersection but it follows from the following simple observation. If A;B;C and D are four distinct points in P and the intersection AB \CD is not empty, then it is either a point (hence a closed set) or a piecewise linear curve which starts and ends at corners of @P(thus being the finite union of closed connected segments). minimal extension for the ˛-manhattan norm 779 Lemma 2.4 ([13, Lemma 2.5]).Let P be a polygon, let A;B2@P be two points such that the segment AB is not contained in @P , then let ı < x ı.P/ and let zAB be a modified geodesic in the sense of Definition 2.2. Let also P1 and P2 be the two polygons in which P is divided by zAB , and let ">0 be a given constant. If ı is small enough, depending only on "and P, then the following is true. For any two points C;D2Pifor i2 ¹1; 2º, one has (2.1) Pi.C;D/<P.C;D/C": If C2P1 , D2P2 and E2@P1\@P2 is any point with distance at most ı from CD, then (2.2) P1.C;E/CP2.E;D/<P.C;D/C": Definition 2.5 (Set of vertices of a geodesic curve).Let PR2 be a polygon. For every A;B2@P , there is a unique ordered set X.A;B/D ¹X1; : : : ; XNº such that the geodesic AB is exactly the piecewise linear curve AX1;:::;XNB , and the points Xj are all the vertices of P met by the geodesic AB (except A and B themselves, in case they are already vertices). The set X.A;B/is called the set of vertices of AB. Definition 2.6 ( ı -linearization of a Jordan curve).Let be a Jordan curve with finite length, and let ı > 0 be much smaller than the diameter of the bounded component of R2n . Let _ A1B1;_ A2B2;:::; _ ANBN be finitely many essentially disjoint arcs contained in . Let then ' be the closed curve obtained by replacing each arc _ AiBi with the segment AiBi. We say that 'is a ı-linearization of if •'is injective, •every arc _ AiBiis such that H1._ AiBi/<ı, •_ AiBi\'AiBi. The ı -linearization is said complete if the union of the arcs _ AiBi is the whole curve ; hence, 'is piecewise linear. Lemma 2.7 ([13, Corollary 4.3]).Let R2 be a convex polygon, and let W@ ! R2 be a parametrized Jordan curve with finite length and let 'W@ !R2 be a ı-linearization of . Then, for every P; Q 2@, one has '.@/'.P /; '.Q/ .@/ .P /; .Q/C2ı: In particular, for every 2Œ0; 2/, we deduce ‰.'/ ‰. / C2ıH1.@/: d. campbell, a. kauranen and e. radici 780 We conclude this section recalling two extension results that will be useful in the sequel. The next proposition (Proposition 2.8) is proved in [13, Theorem A], and Corollary 2.10 is a straightforward consequence of Proposition 2.8. Proposition 2.8 (Minimal extension for standard Manhattan norm).Let RR2 be a rectangle of the form Œa; aCŒb; bC , and let 'W@R!R2 be a continuous injective map. Then, for every ">0 , there exists a piecewise affine homeomorphism vWR!R2coinciding with 'on @Rsuch that kDvk0.R/‰0.'/ C": Moreover, if ' is piecewise linear, then the map v can be chosen finitely piecewise affine. Theorem 2.9 (Minimal extension for standard Manhattan norm).Let ">0 , and let v be the mapping from Proposition 2.8. Then, jD1vj.Q/ZbC b P'.H1 t/; '.H2 t/dt C"; jD2vj.Q/ZaC a P'.V 1 s/; '.V 2 s/ds C": Proof. The finitely piecewise affine homeomorphisms from a rectangle to a polygon in [13] used in the proof of Proposition 2.8 are constructed in [13, Lemma 2.12]. The key estimates we need to extract are the last two unnumbered equations of the proof, found in [13, p. 543]. They say exactly that jD1vj.Q/ZbC b P'.H1 t/; '.H2 t/dt C"; jD2vj.Q/ZaC a P'.V 1 s/; '.V 2 s/ds C": Corollary 2.10 ( W1;1 extension with non-optimal bound).There exists z C > 0 such that the following holds. Let RR2 be a rectangle, let @R be its boundary and let 'W@R!R2 be a continuous, piecewise linear and injective map. Then, there exists a finitely piecewise affine homeomorphism vWR!R2extending 'such that kDvkL1.R/z CH1.@R/H1'.@R/: Proof. The conclusion follows by applying Proposition 2.8 with "DH1.@R/H1'.@R/ minimal extension for the ˛-manhattan norm 787 C S0 P PC P0 x1 H2 t1 H1 t1H1 t1 H2 t1 V1 s V3 s V2 s V3 s V2 s V1 s Figure 3. The figure shows the slicing of the set  into C[S0 by the horizontal line R ¹t1º and Pinto PC[P0by the modified geodesic called x1. number M , where MM1CM2CM3 . We define the strips SiD.RŒti;tiC1/ \ . They are all convex quadrilaterals with two horizontal sides. Step II. Definition of the curve x1D x'. \.R ¹t1º// and the polygons C[S0D and PC[P0DP .The goal of this step is to define the piecewise linear curve x1 , internal to P , which will be the image of the segment H1 t1H2 t1 in a map x' extending ' . The precise parametrization of x'will be presented in the next step; here we only aim to define the curve x1P. Our argument is recursive and so we deal with the first curve x1 defined on H1 t1H2 t1 separating  into S0 and \RŒt1; h DC (see Figure 3). Similarly, the curve x1 means dividing the polygon P into two further polygons: a polygon P0 (which will be the image of S0 ) containing the curve '.H1 t0H2 t0/ and another polygon PC (which will be the image of C) (see Figure 3). Since P is a non-degenerate polygon, let x ı.P/>0 be the parameter of Definition 2.2 and let ı1> 0 be so small that ı1<min ²x ı.P/; ".t1t0/ 8hH1.@P/;h 23;" 2³ and Lemma 2.4 applies with ı1for Pand ".t1t0/ 8h.` Ch/: We define x1as a ı1-modification of the geodesic in Pconnecting H1 t1and H2 t1: Step III. Definition of x' on @S0 .In this step, we care about the definition of x' on @S0 . More precisely, we let x'D'on @ and we specify the parametrization x'W@C\R ¹t1º! x1 so that x'is continuous, injective and piecewise linear, and (3.8) ‰0.x'e@S0/C‰0.x'e@C/‰0.'/ C" 2h.t1t0/: d. campbell, a. kauranen and e. radici 788 Let us observe that thanks to Lemma 2.7 it is enough to look for a continuous and injective parametrization W@C[@S0!R2 coinciding with ' on @ such that (3.8) holds for with error " 4h .t1t0/; namely, ‰0. e@S0/C‰0. e@C/<‰0.'/ C" 4h.t1t0/: Indeed, the correct x' can be found as a ı -linearization of for some ı small enough depending on " 4h .t1t0/such that ‰0.x'e@S0/C‰0.x'e@C/<‰0. e@S0/C‰0. e@C/C" 4h.t1t0/: Thanks to our choice of ı1 and the fact that x1 is a ı1 -modification with variable endpoints of the geodesic connecting H1 t1 and H2 t1 , hence splitting P into the two polygons P0and PC, we can apply Lemma 2.4 to get that (3.9) P0H1 t;H2 tPH1 t;H2 tC".t1t0/ 8h.` Ch/ for any t0< t < t1; PCH1 t;H2 tPH1 t;H2 tC".t1t0/ 8h.` Ch/ for any t1< t < tM: For short, denote c1D.H1 t1/1 and c2D.H2 t1/1 . Then, 0c1< c2` . For every 0 < s < ` , we call s the geodesic inside P connecting V1 s and V2 s . Moreover, whenever c1< s < c2 , we also set V3 sWD .s; t1/ the point in the intersection of H1 t1H2 t1 with V1 sV2 s . For every s2.0; c1/[.c2; `/ , we have that either V1 s; V 2 s2S0 and using Lemma 2.4, P0.V1 s;V2 s/P.V1 s;V2 s/C".t1t0/ 8h.` Ch/; or V1 s; V 2 s2Cand by Lemma 2.4, PC.V1 s;V2 s/P.V1 s;V2 s/C".t1t0/ 8h.` Ch/: The two equations above can be expressed simultaneously as (3.10) max ®P0.V1 s;V2 s/; PC.V1 s;V2 s/¯P.V1 s;V2 s/C".t1t0/ 8h.` Ch/ for all s2.0; c1/[.c2; `/. On the other hand, whenever s2.c1;c2/ , the points V1 s2P0 and V2 s2PC ; thus, the geodesic s necessarily intersects x1 . Let  be the (injective and continuous) constantspeed parametrization of x1 from Œ0; H1.x1/ , .0/ DH1 t1 and .H1.x1// DH2 t1 . For every s2.c1; c2/, we then let X.s/ be the point in s\ x1such that X.s/ Dmax ®x20; H1.x1/W.x/ 2s\ x1¯: minimal extension for the ˛-manhattan norm 789 Then, thanks to Lemma 2.3, it is easy to see that the map s!1.X.s// is nondecreasing; therefore, if c1< s < s0< c2, then X.s0/21.X.s//; H1.x1/: Notice that, in general, the function s!1ıX.s/ is not continuous, nor injective nor surjective. However, any one-dimensional monotone function can be approximated uniformly by strictly monotone functions. Further, it is always possible to slightly modify these strictly monotone approximations in such a way that they become continuous and the price of this is loosing the control on the uniform distance from the original function on a subset whose measure can be made as small as desired. Then, for every  > 0 , it is always possible to find a continuous bijection X of Œc1; c2 onto x1 such that (3.11) H1.J/< where JWD ®s2.c1; c2/WˇˇX.s/ X.s/ˇˇ> ¯: We can then fix Dı1 2and apply Lemma 2.4 to get that (3.12) P0V1 s;Xı1 2 .s/CPCXı1 2 .s/; V2 sP.V1 s;V2 s/C".t1t0/ 8h.` Ch/ for all s2.c1; c2/nJı1 2 . On the other hand, we have the trivial estimate (3.13) P0V1 s;Xı1 2 .s/CPCXı1 2 .s/; V2 sH1.@P/CH1.x1/<2H1.@P/ for all s2Jı1 2 . We define W@C[@S0!R2 as D' on @ and .V 3 s/DXı1 2 .s/ for every s2.c1; c2/ . In particular, is continuous and injective and fails to be piecewise linear only on the segment H1 t1H2 t1 . Moreover, gathering together (3.9) , (3.10) , (3.12) and (3.13), we deduce that ‰0. e@S0/C‰0. e@C/ DZt1 0 P0 .H1 t/; .H2 t/dt CZh t1 PC .H1 t/; .H2 t/dt 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. kauranen and e. radici 790 Zh 0 P'.H1 t/; '.H2 t/dt CZ.0;`/nJı1 2 P'.V 1 s/; '.V 2 s/ds C".t1t0/ 8h.` Ch/`ChH1Jı1 2CH1Jı1 22H1.@P/ ‰0.'/ C" 4h.t1t0/; where in the last inequality we used (3.11) and the fact that ı1<".t1t0/ 8hH1.@P/. Finally, thanks to Lemma 2.7 and the considerations of the first part of the step, we can find a function x'W@C[@S0!R2 that is continuous, injective and piecewise linear and such that (3.8) holds. Step IV. Definition of z' on @T 1 0[@R0[@T 2 0 .In this step, we further subdivide the strip S0 in the essentially disjoint union of two triangles T1 0 , T2 0 and a rectangle R0 with the following properties. The rectangle R0 is the biggest rectangle with horizontal and vertical sides inside S0 , such that the horizontal sides are contained in @S0 , while T1 0,T2 0are the two disjoint right-angle triangles containing I1 0,I2 0, respectively. We continue to define some new z'W@C[@T 1 0[@R0[@T 2 0!R2 coinciding with x' on @C[@S0 such that z' is injective, continuous and piecewise linear and satisfies the following estimate: (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.t1t0/: Let us emphasize that z'will be defined so that z'.@T 1 0[@R0[@T 2 0/P0. We denote by d1 and d2 the two values such that the projection of @S0 onto R ¹0º is exactly Œd1;d2 ¹0º , and we call x1 and x2 those values for which the projection of @R0 onto R ¹0º is the segment Œx1; x2 ¹0º . Notice that d1c1x1< x2c2d2 . Since P0 is a non-degenerate polygon, we let x ı.P0/ be the parameter of Definition 2.2 and take ı0 1<min ²x ı.P0/; ".t1t0/ 32hH1.@P0/³ and Lemma 2.4 applies with ı0 1for P0and ".t1t0/ 16h.`Ch/ . Let now x1 be the geodesic inside P0 connecting x'.V 1 x1/ and x'.V 3 x1/ and let xx1 be its ı0 1 -modification in the sense of Definition 2.2. In particular, xx1 splits P0 into two non-degenerate polygons P1 0 and U , where P1 0 contains x'.I 1 0/ and U contains x'.I 2 0/ , where I1;2 0 are the two non-horizontal segments of @S0\@ . This situation is depicted in Figure 4. minimal extension for the ˛-manhattan norm 791 20 d. campbell, a. kauranen and e. radici 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) Figure 4. The sets 𝑇1 0, 𝑇2 0and 𝑅0, and P1 0and Uin Step IV. while for the remaining 𝑡∈𝐽𝑙 𝛿′ 1 2 we get (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) give 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 step III, Lemma 2.7 ensures that there is some continuous, injective and piecewise linear map ˜𝜑1:𝜕𝑇1 0∪𝜕𝑆0→R2such that ˜𝜑1=𝜓1=¯𝜑on 𝜕𝑆0and Ψ0(˜𝜑1 ⌉𝜕𝑇1 0 ) + Ψ0(˜𝜑1 ⌉𝜕(𝑆0\𝑇1 0)) ≤ Ψ0(¯𝜑⌉ P0) + 𝜀 4ℎ(𝑡1−𝑡0). To conclude the step we need to repeat the very same argument on ˜𝜑1 ⌉𝜕(𝑆0\𝑇1 0) by replacing P0 with U and considering a 𝛿′′ 1 -modification of 𝜈𝑥2 , where 𝛿′′ 1 is chosen so Figure 4. The sets T1 0,T2 0,R0,P1 0and Uin Step IV. Thanks to Lemma 2.4, we have (3.15) Ux'.V 1 s/; x'.V 3 s/P0x'.V 1 s/; x'.V 3 s/C".t1t0/ 16h.`Ch/;for all s2.x1; x2/; Ux'.V 1 s/; x'.V 2 s/P0x'.V 1 s/; x'.V 2 s/C".t1t0/ 16h.`Ch/;for all s2.x2; d2/; while for 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".t1t0/ 16h.` Ch/: We continue similarly as described in Step III. For every t2.0; t1/ , we denote the point x'.H1 t/DH1 t2P1 0 and x'.H2 t/DH2 t2U ; thus, the geodesic t connecting H1 t and H2 t inside P0 must intersect xx1 . So also in this case, for every t2.0; t1/ , we can find a map Y.t/ identifying the last point of the intersection t\ xx1 running xx1 from x'.V 1 x1/ to x'.V 3 x1/ . Moreover, exactly as explained in Step III, we can find a continuous and injective approximation Yı0 1 2 such that ˇˇY.t/ Yı0 1 2 .t/ˇˇ<ı0 1 2for all t2.0; t1/nJ1 ı0 1 2 and H1J1 ı0 1 2<ı0 1 2: So if we now call H3 tWD .t; x1/ , then we can define 1W@T 1 0[@S0!R2 in this way: 1D x'on @S0and 1.H3 t/DYı0 1 2 .t/ for all t2.0; t1/: Then, the map 1 is continuous and injective and fails to be piecewise linear only on the segment ¹x1º  Œ0; t1 . Using Lemma 2.4, for all t2.0; t1/nJ1 ı0 1 2 , we can estimate P1 0 1.H1 t/; 1.H3 t/CU 1.H3 t/; 1.H2 t/ P0x'.H1 t/; x'.H2 t/C".t1t0/ 16h.` Ch/; (3.17) d. campbell, a. kauranen and e. radici 792 while for the remaining t2Jl ı0 1 2 , we get P1 0 1.H1 t/; 1.H3 t/CU 1.H3 t/; 1.H2 t/ H1.xx1/CH1.@P0/2H1.@P0/: (3.18) Then, (3.15), (3.16), (3.17) and (3.18) give ‰0 1 e@T 1 0C‰0 1 [email protected] 0/ Zc1 d1 P0x'.V 1 s/; x'.V 2 s/C".t1t0/ 16h.` Ch/ds CZc2 c1 P0x'.V 1 s/; x'.V 3 s/C".t1t0/ 16h.` Ch/ds CZd2 c2 P0x'.V 1 s/; x'.V 2 s/C".t1t0/ 16h.` Ch/ds CZ.0;t1/nJ1 ı0 1 2 P0x'.H1 t/; x'.H2 t/C".t1t0/ 16h.` Ch/dt C2H1J1 ı0 1 2H1.@P0/ ‰0.x'eP0/C" 8h.t1t0/; and, as in Step III, Lemma 2.7 ensures that there is some continuous, injective and piecewise linear map z'1W@T 1 0[@S0!R2such that 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.t1t0/: To conclude the step, we need to repeat the very same argument on z'1 [email protected] 0/ by replacing P0 with U and considering a ı00 1 -modification of x2 , where ı00 1 is chosen so that ı00 1<min ²x ı.U/; ".t1t0/ 32`H1.@U/³ and Lemma 2.4 applies with ı00 1for Uand ".t1t0/ 16h.`Ch/ . This would provide a continuous, injective and piecewise linear map z'W@T 1 0[ @R0[@T 2 0!R2extending z'1(hence, ultimately, x') such that ‰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.t1t0/ ‰0.x'eP0/C" 2h.t1t0/ thus proving (3.14) and concluding the step. minimal extension for the ˛-manhattan norm 793 Step V. Recursion and conclusion. In this final step, we want to conclude our construction by recursion. In Steps III and IV, we divided  into a new convex polygon with two horizontal sides C and a horizontal strip S0 given by a rectangle R0 and two triangles T1 0,T2 0. Then, we defined continuous, injective and piecewise linear functions x'; z' such that x'D z'on @Csatisfies (by (3.8), (3.14) and our choice of ı1) ‰0.x'e@S0/C‰0.x'e@C/‰0.'/ C" 2h.t1t0C2ı1/ ‰0.'/ C" 2h.t1t0/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.t1t0C2ı1/ ‰0.x'e@S0/C" 2h.t1t0/C" 4 1 2: Iterating the construction and choosing for every i the parameter ıi suitably small depending on ti1,h 2iC1and the polygon PCDPnSi1 jD0Pj, we then find M1 X iD0 ‰0.x'e@Si/‰0.'/ C" 2h M1 X iD0 .tiC1ti/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 .tiC1ti/C" 4 M1 X iD0 1 2i; which finally imply (3.6) and (3.7), respectively. 4. Piecewise affine extension In this section, we investigate two possible finitely piecewise affine homeomorphic extension inside triangles. Lemma 4.1 (Extension-direct).Let TR2 be a triangle of corners A; B; C such that BC is horizontal and A is the intersection of BC with the bisector of the angle at A . If 'W@T !R2 is continuous, injective and linear on each of the segments AB; AC; BA and AC , then there exists a bi-affine homeomorphism vWT!R2 such that vD' on @T and kDvk0.T / H1'.@T /H1.@T /: d. campbell, a. kauranen and e. radici 794 Proof. The proof is immediate, and indeed it is enough to consider the continuous map vwhich is affine on each of the triangles T1WD ABA,T2WD BAC. Then, we can compute D1veT1D'.A/'.B/ .A/1.B/1 L2; D1veT2D'.C / '.A/ .C /1.A/1 L2 and D2veT1D'.A/ '.A/D1veT1.A/1.A/1 .A/2.A/2 L2; D2veT2D'.A/ '.A/D1veT2.A/1.A/1 .A/2.A/2 L2: In particular, one can estimate kDvk0.T / D jD1vj.T1/C jD2vj.T1/C jD1vj.T2/C jD2vj.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 (Extension-indirect).Let TR2 be a triangle of corners A; B; C such that AB is horizontal, BC is vertical and let 'W@T !R2 be a continuous, piecewise linear, injective map such that ' is linear on the hypotenuse AC . For every ">0 , there exists a finitely piecewise affine homeomorphism vWT!R2 such that vD' on @T and kDvk0.T / ‰0.'/ C242H1.@T /H1'.AC /C": Proof. Let ">0be fixed arbitrary small. For simplicity of notation, through the proof we refer to ˇ as the internal angle of the corner AD.0; 0/ and we will denote dWD j'.A/ '.C /j D H1.'.AC // and dD jACj. Clearly, since the internal angle in Bis =2, then ˇ2.0; =2/. minimal extension for the ˛-manhattan norm 795 X Y Z A B C '.A/ '.C / D E F z T G H1 tH2 tH3 t V3 s V2 s V1 s PADEC P Pz T Q  z'.D/'.E/ P '.D/ '.F / ˇ Figure 5. The decomposition of T into TD[z T[ADEC , and the corresponding decomposition of int '.@T / into P[Pz T[PADEC . Since the polygon of boundary '.@T / is non-degenerate, then Definition 2.2 provides some constant x ı > 0, and then we consider (4.1)  < ²1; d 2; x ı 4;d 14;" 4;1C1 2tan ˇ1 ;kD'k1 1³: The basic idea of the proof is to find a suitable one-dimensional skeleton ‡ inside T , construct a continuous, piecewise linear and injective map z'W‡!R2 coinciding with ' on @T and finally perform a suitable piecewise affine extension inside each component of the partition of Tidentified by ‡. For clarity, we present the proof in three separate steps. Step I. Definition of a first skeleton „ and a continuous piecewise linear injective map '1W„!R2 .In this step, we would like to construct a one-dimensional skeleton of the form „D@T [DE [F G , for some suitably chosen points D; E; F; G , and we will define an extension '1 of ' on DE [F G that is still continuous, piecewise linear and injective. See Figure 5for an illustration. We will select D; E; F; G so that D2AB; E; F 2BC; G 2DE; F G kAB and DE kAC; satisfying the following estimates: (4.2) jADj<  and jCEj< ; H1'1.@ z T /< 4; ‰0.'1 e@/‰0.'/ C.dC14/H1.@T /; where z TT is the triangle of corners E; F; G and T is the trapezoid of corners D; B; F; G. d. campbell, a. kauranen and e. radici 796 Having fixed  , by the assumptions on ' , we can choose D2AB and E2BC so that (i) jADj<  and jCEj< ; (ii) j'.A/ '.D/j<  and j'.C / '.E/j< ; (iii) the restriction of 'is linear on AD and CE; (iv) DE is parallel to AC ; (v) the point X is on the internal bisector of '.A/ and Y on the internal bisector of '.C / such that j'.A/ Xj< 2 and j'.C / Yj< 2 and the piecewise linear path z'.D/'.E/ WD '.D/XY'.E/ lies in the interior of '.@T / and is a x ı=2-modification of the geodesic '.D/'.E/ in the sense of Definition 2.2. Observe that (ii) and (v) imply that 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 find a point F on the segment EB , a point G2DE and a point Z2ŒY'.E/ such that (vi) jFEj<  and j'.F / '.E/j< ; (vii) 'is linear on EF ; (viii) j'.E/ Zj<  and Œ'.F /Zlies in int '.@T /, and as a consequence, j'.F / Zj< 2; (ix) G2DF2,jGEj< .sin ˇ/1and jGFj< .tan ˇ/1. This concludes the definition of „ . Indeed, (i) ensures the first equation of (4.2) , then z Tis a right-angle triangle and is a trapezoid inside T. We now proceed to construct a function '1W„!R2 extending ' such that the second and third estimates of (4.2) are satisfied. In order to do that, we consider two further auxiliary points P; Q 2DG so that (x) jPDj<  and jQGj< ; and we set '1.P / WD X; '1.Q/ WD Yand '1.G/ WD Z: We then define '1W„!R2 so that '1D' on @T , '1 eDP is the parametrization at constant speed of the segment '.D/X , '1 ePQ is the parametrization at constant speed of the segment XY , '1 eQG is the parametrization at constant speed of the segment YZ , '1 eGE is the parametrization at constant speed of the segment Z'.E/ and, finally, '1 eGF is the parametrization at constant speed of the segment Z'.F / . A sketch of the situation is presented in Figure 5. minimal extension for the ˛-manhattan norm 803 We observe that v is continuous and injective, hence a homeomorphism, since vD z' on ‡ . From the same observation, we also deduce that vD' on @T because z'D' there. Gathering together (4.10), (4.11), (4.12) and (4.7), we find kDvk0.T / D kD!k0.P/C kDwk0.z T / C M1 X iD0 kDvik0.Ri/C kDvk0.y T / z C ddC21C1 2tan ˇC M1 X iD0‰0.z'e@Ri/C" 2iCd" z C ddC21C1 2tan ˇC‰0.'/ CCdH1.@T / Cz C " ‰0.'/ Cz CH1.@T /H1'.AC /Cz C "; where in the last inequality, we used that dD jACj  H1.@T / and dDˇˇ'A'.C /ˇˇ: 5. Proof of Theorems 1.1 and 1.2 Proof of Theorem 1.1.Let ">0be arbitrary fixed and let (5.1) 0 <  < min ²1; " z CCz CH1.Q/³ for some large appropriate but fixed geometric constant z C . We describe in detail the proof when Q is of class (iii) in the sense of Remark 3.1, while the other cases are an obvious modification of the current argument. Applying Lemma 3.2 to Q; '; ˛ and the parameter  , we can partition Q in two triangles T1; T2 and a convex polygon  and find a continuous, piecewise linear, injective map x'W@T1[@T2[@ !R2 with the properties listed in the statement of Lemma 3.2. In particular, from (3.3), it follows that (5.2) ‰˛.x'e@/‰˛.'/ C; where (3.1) and (3.2) ensure that (5.3) H1.@T1/CH1.@T2/ < ; H1x'.@T1/CH1x'.@T2/< : Furthermore, since  is a convex polygon with two parallel sides in direction ˛ , we can apply the ˛ -rotated version of Lemma 3.3 to  , x'1 and the parameter  to find M increasing values .ti/M1 iD0 and ˛ -rotated strips Si , which can be seen as the union of a d. campbell, a. kauranen and e. radici 804 rectangle Ri and two triangles T1 i and T2 i , and a continuous piecewise linear injective map O'WSM1 iD0@T 1 i[@Ri[@T 2 i!R2 coinciding with x' on @ with the properties of Lemma 3.3. In particular, thanks to (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) ensures that (5.5) H1O'.I 1 i/CH1O'.I 2 i/<  where I1 iD@T 1 i\@ and I2 iD@T 2 i\@. We are finally in position to define a function z'W@T1[ M1 [ iD0 @T 1 i[@Ri[@T 2 i![@T2!R2 that is continuous, injective, finitely piecewise linear and such that z'D x'1 on @T1[@T2 and z'D O'on SM1 iD0@T 1 i[@Ri[@T 2 i. Once here we will perform the homeomorphic piecewise affine extension on T1; T2; T 1 i; T 2 i and Ri independently for every iD0; : : : ; M 1 . We first focus on the extension inside the strips Si . Let i2 ¹0;:::;M 1º be fixed; we then apply the ˛ -rotated version of Proposition 2.8 to Ri;z'e@Ri and parameter .tiC1ti/ to find finitely piecewise affine homeomorphisms viWRi!R2 coinciding with z' on @Ri such that (5.6) kDvik˛.Ri/‰˛.z'e@Ri/C.tiC1ti/: By construction, we have that T1 i; T 2 i are right-angle triangles whose hypotenuse is contained in @ \@Q , and we can apply the ˛ -rotated version of Lemma 4.2 to T1;2 i , z'e@T 1;2 i and parameter .tiC1ti/to find finitely piecewise affine homeomorphisms w1;2 iWT1;2 i!R2 coinciding with z'on @T 1;2 isuch that 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 .tiC1ti/; which thanks to (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 .tiC1ti/: (5.7) minimal extension for the ˛-manhattan norm 805 Let us also notice that, for future need, since the triangles T1;2 i have one angle equal to =2 and, by construction, their hypotenuse is I1;2 i , then one has H1.@T 1;2 i/ 3H1.I1;2 i/ for every i . Moreover, having I1;2 i.@ \@Q/ and the triangles pairwise essentially disjoint, 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/: Let us now consider the extension inside the triangles T1; T2 . In this case, by construction, we are in position to apply the ˛ -rotated version of Lemma 4.1 to T1;2 and z'e@T1;2 to find bi-affine homeomorphisms w1;2 WT1;2 !R2such that kDw1;2k˛.T1;2/H1z'.@T1;2/H1@T1;2 which, thanks to (5.3), gives (5.9) kDw1k˛.T1/C kDw2k˛.T2/ < 22: We can finally define vWQ!R2to be the piecewise affine function such that vDw1;2 on T1;2; v Dw1;2 ion T1;2 iand vDvion Rifor every iD0;: : :;M 1: By construction, v is continuous because vD z' on the one-dimensional skeleton @T1[.SM1 iD0@T 1 i[@Ri[@T 2 i/[@T2 , and, moreover, v coincides with z'D' on @Q . To conclude, it is only left to verify the validity of (1.4) , but this is now a straightforward consequence of (5.6) , (5.7) , (5.9) , (5.4) , (5.8) and (5.2) . Indeed, we get kDvk˛.Q/D kDw1k˛.T1/C kDw2k˛.T2/ C M1 X iD0kDw1 ik˛.T 1 i/C kDvik˛.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.tiC1ti/ ‰˛.x'1 e@/CC42Cz C H1.@Q/Cdiam Q ‰˛.x'e@/C6 Cz C H1.@Q/; ‰˛.'/ Cz C 1CH1.@Q/; and then estimate (1.4) follows since has been chosen as in (5.1). d. campbell, a. kauranen and e. radici 806 5.1. Proof of Theorem 1.2 To prove the claim, it suffices to repeat the proof of the above lemmas as before but using the estimates from Theorem 2.9 instead of from Proposition 2.8. In all of our calculations, we estimate RP.'1.H2 t/;'1.H 3 t//dt and RB1 D1P.'1.V 1 s/;'1.V 2 s//ds separately. Now it suffices to keep them separate instead of summing them. The key estimates in Lemma 3.2 are (3.4) , in Lemma 3.3 they are (3.12) , (3.13) and the calculation following, and in Lemma 4.2 they are (4.4), (4.5), (4.6). 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Zbl 1359.46036 MR 3519110 Received 20 January 2023, and in revised form 30 August 2023 Daniel Campbell Department of Mathematical Analysis, Charles University Sokolovská 83, 186 00 Prague 8, Czech Republic [email protected] Aapo Kauranen Department of Mathematics and Statistics, University of Jyväskylä PL 35, 40014 Jyväsklyän yliopisto, Finland [email protected] Emanuela Radici Department of Information Engineering, Computer Science and Mathematics (DISIM), University of L’Aquila Via Vetoio 1 (Coppito), 67100 L’Aquila, Italy [email protected]