APPROXIMATION BY BV-EXTENSION SETS VIA PERIMETER MINIMIZATION IN METRIC SPACES

. We show that every bounded domain in a metric measure space can be approximated in measure from inside by closed BV -extension sets. The extension sets are obtained by minimizing the sum of the perimeter and the measure of the diﬀerence between the domain and the set. By earlier results, in PI-spaces the minimizers have open representatives with locally quasiminimal surface. We give an example in a PI-space showing that the open representative of the minimizer need not be a BV -extension domain nor locally John.


Introduction
In this paper we study the existence of BV -extension sets in complete and separable metric measure spaces X.By BV -extension sets we mean sets E for which any integrable function with finite total variation on E can be extended to the whole space X without increasing the BV -norm by more than a constant factor.BV -and Sobolev-extension sets are useful in analysis because via the extension one can use tools a priori available only for globally defined functions also for the functions defined only in the extension set.Not every domain of a space is an extension set, so in cases where one starts with functions defined on an arbitrary domain Ω one first approximates Ω from inside by an extension set, then restricts the functions to this set and then extends them as global functions.Such process immediately raises the question: when can we approximate a domain from inside by extension domains (or sets)?
In the Euclidean setting, an answer to this has been known for a long time.For instance, from the works of Calderón and Stein [7,20] we know that Lipschitz domains of R n are W 1,p -extension domains for every p ≥ 1.Any bounded domain in R n can be easily approximated from inside and outside by Lipschitz domains.It was later observed that in a more abstract setting of PI-spaces (that is doubling metric measure spaces satisfying a local Poincaré inequality [14]; see Section 4), good replacements of Lipschitz domains are uniform domains.In [4] it was shown that uniform domains in p-PI-spaces are N 1,p -extension domains, for 1 ≤ p < ∞, for the Newtonian Sobolev spaces, and in [17] it was shown that bounded uniform domains in 1-PI-spaces are BV-extension domains.Finally, in [19] it was shown that in doubling quasiconvex metric spaces one can approximate domains from inside and outside by uniform domains.Since PI-spaces are quasiconvex [9,15], we conclude that in PI-spaces one can approximate domains by extension domains.
Recently there has been increasing interest in analysis in metric measure spaces (X, d, m) without the PI-assumption.However, the extendability of BV -functions seems to have been studied only in Date: May 5, 2023.2000 Mathematics Subject Classification.Primary 30L99.Secondary 46E35, 26B30.
1 some specific cases, such as infinite dimensional Gaussian case [5].We continue into the direction of general metric measure spaces and show in Theorem 3.4 that even without the PI-assumption one can still approximate domains Ω from inside by closed BV -extension sets.It is not clear if an approach similar to the approximation by uniform domains could work in general metric measure spaces.Therefore, we take a completely different approach and obtain the extension set by minimizing the functional A → Per(A) + λm(Ω \ A) for a large parameter λ > 0. Section 3 contains the proof of Theorem 3.4 and remarks on the minimization procedure.Before it, in Section 2 we recall and prove preliminary results on BV -functions and sets of finite perimeter.In Section 4 we connect the minimization approach to domains with locally quasiminimal boundary in PI-spaces, and also show that in PI-spaces the open representatives of the minimizers of the functional, and consequently domains with locally quasiminimal boundary need not be BV -extension domains, nor locally John domains.In the final part of the paper, Section 5 we list open questions raised by our extension result.

Preliminaries
We will always assume (X, d, m) to be a metric measure space where (X, d) is a complete and separable metric space and m is a Borel measure that is finite on bounded sets.The set of all Borel subsets of X is denoted by B(X).We define the open and the closed ball with center x ∈ X and radius r > 0 by B r (x) := {y ∈ X : d(x, y) < r} and Br (x) := {y ∈ X : d(x, y) ≤ r}, respectively.We shall denote by LIP(X) the space of all Lipschitz functions on X and by Lip(f ) the (global) Lipschitz constant of f ∈ LIP(X).Given any f ∈ LIP(X) and E ⊂ X we set Lip(f ; Having this notation at our disposal, the asymptotic Lipschitz constant (or the asymptotic slope) of a function f ∈ LIP(X) is a function lip a (f ) : X → [0, +∞) given by Notice also that lip a (f ) ≤ Lip(f ).Given an open set A ⊂ X we will say that a function f : X → R is locally Lipschitz on A if for every x ∈ A there exists r > 0 such that B r (x) ⊆ A and f | Br(x) is Lipschitz.We denote the space of all locally Lipschitz functions on A by LIP loc (A).
Functions of bounded variation.We next recall the definition of the space of functions of bounded variation (BV functions, for short), as well as some of the characterisations of the total variation (measure) associated with a BV function.The below presentation is based on [11].
We extend |Df | X to all Borel sets as follows: given B ∈ B(X), we define 18,Thm. 3.4]).It follows from the definition that, given an open set A ⊂ X Given a Borel set B ⊂ X and f ∈ L 1 loc (m | B ), we introduce the following notation: |Df | B := the total variation measure of f computed in the metric measure space (X, d, m | B ).

Definition 2.2 (The spaces
• BV (B) and BV (B)).Let (X, d, m) be a metric measure space.Let B ⊂ X be Borel.We define We endow the space • BV (B) with the seminorm and the space BV (B) with the norm given by Remark 2.3.The following characterisation of the total variation measure of the whole space will be useful for our purposes.By [11,Theorem 4.5.3]we have that In general, we cannot restrict to globally Lipschitz functions when calculating the total variation measure: consider A = (0, 1) ∪ (1, 2) ⊂ R and f = χ (0,1) .
We will use the following version of Lipschitz extensions where the asymptotic Lipschitz constant is preserved.Proposition 2.4 ([12, Theorem 1.1]).Let (X, d) be a metric space, C ⊂ X a subset and g : C → R a Lipschitz function.Then for every ε > 0 there exists an (Lip(g) + ε)-Lipschitz function f : X → R whose restriction to C coincides with g and such that for every x ∈ C.
Moreover if g is bounded (resp.with bounded support), then f can be chosen to be bounded (resp.with bounded support).
By combining Proposition 2.4 with Remark 2.3 we get the following.
Proof.By Proposition 2.4 every f ∈ LIP(B) can be extended to an element of LIP(X) without changing the asymptotic Lipschitz constant on B, thus (taking into account Remark 2.3) we obtain and thus BV (B) = BV (Y ) (cf. [12,Theorem 3.1]).Now, take A ⊂ X open.Since every f ∈ LIP loc (A) can be restricted to an element of LIP loc (B ∩A), we get that By the definition of total variation measure, the inequality (3) extends to all Borel sets A ⊂ X.Finally, by (2) and recalling that |Df | Z is a finite Borel measure for any metric measure space (Z, d Z , m Z ), we have for all Borel A ⊂ X that The equality (1) follows by taking A = B in the above equality, combined with Remark 2.3 and Proposition 2.4.
We define the notion of sets of finite perimeter on a Borel subset B ⊂ X. Definition 2.6 (Sets of finite perimeter on a Borel subset B).Let (X, d, m) be a metric measure space and B, E ∈ B(X).We define the perimeter of E on B as We say that E has finite perimeter on B if the quantity Per B (E) is finite.Moreover, we define for every F ∈ B(X) the quantity Per B (E; To shorten the notation, whenever B is equal to the whole (base) space X, we will often write Per(E) instead of Per X (E).
Extension sets and extension properties.Definition 2.7 (BV -extension set).A set B ∈ B(X) is said to be a BV -extension set if there exist C > 0 and a map E B : BV (B) → BV (X), such that for every f ∈ BV (B) the following hold: i Given a BV -extension set B, we define the operator norm of E B as Definition 2.8 (Extension property for sets of finite perimeter).Let B ∈ B(X).We say that B has the extension property for sets of finite perimeter with respect to the full BV -norm if there exists C > 0 such that for every E ⊂ B with Per B (E) < +∞ there exists E ∈ B(X) such that the following two properties hold:

Approximation by BV-extension sets from inside
In this section we prove the main result of the paper, Theorem 3.4, according to which we can estimate domains from inside by closed BV -extension sets.In the proof we will need the following two results.The first one connects the extendability of BV -functions with the extendability of sets of finite perimeter.In the Euclidean case, such result was obtained by Burago and Maz'ya [6].Later it was extended to PI-spaces by Baldi and Montefalcone [3].The connection of perimeter-and BVextensions with W 1,1 -extensions was studied in detail in [13] in Euclidean spaces, and then in general metric measure spaces in [8].In [8,Proposition 3.4] the extension result closest to what we need was proven.There a Borel set was shown to be a BV -extension set if and only if it has the extension property for Borel sets of finite perimeter with the full norm.We need to make a small modification to this result, since in our proof we need to stay in the class of closed sets and consequently will only use open sets for testing the perimeter extensions.Proposition 3.1.Let (X, d, m) be a metric measure space.A Borel subset Ω ⊂ X has the extension property for BV if and only if it has the extension property for open sets of finite perimeter with the full norm.
Proof.Having already the equivalence between BV -extension and perimeter extension of Borel sets given by [8, Proposition 3.4], we only need to show that perimeter extension for open sets implies BVextension for functions in BV (Ω)∩L ∞ (Ω).Towards this, take f ∈ BV (Ω)∩L ∞ (Ω).By the definition of the total variation, there exists a sequence of open sets U n ⊃ Ω and functions

Now, by assumption we can extend each relatively open set
, where C > 0 is the constant given by the assumption on having the extension property for open sets.
As in the proof of [8,Proposition 3.4], this implies that we get an extension fn ∈ BV (X) of .
By an application of Mazur's lemma (see again the proof of [8, Proposition 3.4] for details), this implies that we also get an extension This concludes the proof.
The next lemma is the reason why our approach works only for closed sets.Later in Example 3.3 we observe that the claim of the lemma fails for general sets B ⊂ X. Lemma 3.2.Let (X, d, m) be a metric measure space.Given a closed set B ⊂ X and a set A ⊂ B of finite perimeter on B, it holds that Since B is closed, by Corollary 2.5 there exists a sequence (g i ) i ⊆ LIP(X) such that (5) For a fixed i ∈ N we then have that Therefore, up to taking a (relabeled) subsequence of (f i ) i , we may assume that ( 6) Taking into account (6), this gives where the last inequality follows from ( 5) and ( 4).
Notice that Lemma 3.2 does not hold in general if we replace the closed set B with a general Borel set.This is seen from the next simple example.
Example 3.3.Let us consider (R, d Eucl , L 1 ) as our metric measure space.Let B = (0, 1) ∪ (1, 2) and A = (0, 1).Then we have that Theorem 3.4.Let (X, d, m) be a metric measure space.Let Ω ⊂ X be a bounded open set.Then for every ε > 0 there exists a closed set G ⊂ Ω such that m(Ω \ G) < ε and so that the zero extension gives a bounded operator from BV (G) to BV (X).
Proof.Let us denote C Ω = {A ⊂ Ω : A closed}.We consider the following functionals.For λ > 0 define M λ : C Ω → [0, +∞] as M λ (A) := Per(A) + λm(Ω \ A).We will show that for λ large enough, a minimal element in a partial order given by M λ will give the desired set G. We divide the proof into several steps.
Step 1: For every λ > 0, we have inf A∈C Ω M λ (A) < +∞.Moreover, given ε > 0 there exists λ > 0 such that for any sequence Together with the above, this gives the existence of s ∈ [0, r] such that proving the first part of the claim.Let now ε > 0. Take r > 0 so small that m r = m(B(∂Ω, r)∩Ω) < ε 2 .Note that for any λ > 0, we have lim i→∞ M λ (A λ i ) ≤ mr r + λm r and so, by taking λ > 1 r , we get This proves the claim of Step 1.
Next, we shall consider the following (non-empty, due to Step 1) subset of C Ω : Consider now a partial order A ≺ λ B on C Ω,λ defined as Step 2: For every λ > 0 and C ∈ C Ω,λ , the set {A ∈ C Ω,λ : A ≺ λ C} has a minimal element with respect to the partial order ≺ λ .
Proof of Step 2. By Zorn's Lemma, it suffices to prove that any chain (A λ i ) i∈I ⊂ {A ∈ C Ω,λ : A ≺ λ C} contains a lower bound.By selecting inductively elements in the chain so that m(A λ i \ A λ j ) > 0, we may assume that I = N.Moreover, we may assume that A λ i+1 ⊂ A λ i for all i ∈ N. We claim that gives the lower bound.Trivially, A λ ⊂ A λ i for all i ∈ N, so it is enough to prove that M λ (A λ ) ≤ M λ (A λ i ) for all i ∈ N. To verify the latter, notice that by the continuity of measure, we have that and so by the lower semicontinuity of the perimeter, we have also Per(A λ ) ≤ lim inf i→+∞ Per(A λ i ), proving the claim.We now show that for any λ > 0 and a minimal element G λ ∈ C Ω,λ with respect to ≺ λ we have that the zero extension from G λ gives a bounded operator.Given any Borel set B ⊂ X, in what follows we will denote by E B the zero-extension operator from BV (B) to BV (X).
We are now ready to combine the results obtained in the three steps above and get the claim of the theorem.
Step 4. Fix ε > 0. There exists a closed set G ⊂ Ω such that Proof of Step 4. Let λ (depending on ε) be such that the claim of Step 1 holds and fix any minimizing sequence (A λ i ) i∈N .Then, for i ∈ N large enough we have that i } with respect to the partial order ≺ λ , whose existence has been proved in Step 2. By Step 3 we know that G λ,A λ i is a BV -extension set, thus it only remains to check that m(Ω \ G λ,A λ i ) < ε.To verify this, notice that, by the minimality This proves the statement of Step 4 (and of the theorem itself) for G = G λ,A λ i .
By approximating a measurable set from outside by an open set, Theorem 3.4 gives the following corollary.
Corollary 3.5.Let (X, d, m) be a metric measure space and let F ⊂ X be a bounded Borel set.Then for every ε > 0 there exists a closed set G ⊂ X such that m(F ∆G) < ε and so that the zero extension gives a bounded operator from BV (G) to BV (X).Remark 3.6.A stronger version of Corollary 3.5 where we require in addition that G ⊂ F , does not hold.A counter example is given by taking F to be a fat Cantor set in R equipped with the Lebesgue measure.
We end this section with an example where the set G does not have an open representative.
Example 3.7.Let X = R 2 with the Euclidean distance.We define Ω = Q ∪ ∞ n=1 T n , where Q = (0, 1) × (−1, 0) and T n are defined as follows.We start by defining a triangle with unit length base: Notice that T contains the base, but not the other two sides of the triangle.We then define Furthermore, define where x n = (2 −2n+1 + 2 −2n , 0) is the center point of the base of the triangle T n .
Step 1: Let us show that we can split the functional M λ with respect to the cube Q and the triangles T n .First notice that for all A ⊂ Ω we have Towards showing that the perimeter part of the functional M λ also splits, we next show that for a finite perimeter set is the closed upper half plane.We do this by showing the chain of inequalities The equality in the chain (9) follows by subadditivity.We first show the inequality Per(A∩R 2 + ; R 2 + ) ≤ Per(A; R 2 + ).To this end we define and call U i the 1 i -neighborhood of R 2 + .This way we obtain Further let f i ∈ LIP loc (U i ) be such that f i ⇀ χ A and R 2 lip a (f i ) dm → Per(A; R 2 + ).We may assume that f i have values in [0, 1].Now setting g i = f i φ i we have g i ⇀ χ A∩R 2 + and g i is an admissible sequence of Lipschitz functions for Per(A ∩ R 2 + ; R 2 + ).By the Leibniz rule we now obtain

Next we show the second inequality
To this end we let φ i be as before.Further let We may again assume that f i and g i have values in [0, 1].Therefore, we can set h i = φ i g i + (1 − φ i )f i , for which it holds h i ⇀ χ A .Now again by a similar approximation as before using the Leibniz rule we obtain from which the claimed inequality follows.Now Per + is an open set, we have ).Let us recall that the perimeter measure enjoys the following locality property: given an open set U ⊂ X and sets of finite perimeter E, F ⊂ X such that m(U ∩ (E∆F )) = 0, it holds that (10) Per(E; U ) = Per(F ; U ).
Taking into account that T n are pairwise disjoint compact sets together with (10), one can easily verify that

Consequently, we get
Step 2: Let G λ be a minimizer of M λ .We look to show that for large λ > 0 and n > 0, G λ will contain one of the points x n , but nothing of the respective triangle T n , in the measure sense, i.e. m(G λ ∩ T n ) = 0.This means that G λ does not have an open representative.
Although it is not strictly needed in the following, we first notice that Q ⊂ G λ as long as λ > 0 is large enough.Next we will perform a reflection of the part of G λ that lies inside the triangles T n across the line where Per euc denotes the Euclidean perimeter.The first inequality follows since an admissible Lipschitz function for the definition of the perimeter on the left hand side will define an admissible Lipschitz function for the definition of the Euclidean perimeter on the right hand side via a reflection.For the second inequality we used the Euclidean isoperimetric inequality.Now given λ > 0 and as long as the minimizer G λ contains boundary of positive measure and thus there is no open representative of G λ .
Notice that the example above has a closed representative since we can always add in the boundary of the set G λ to itself since the dirac masses do not add perimeter, and the rest of the boundary is a null set with respect to m.

Remarks on quasiminimal sets in PI-spaces
As noted in the Introduction, in PI-spaces we can approximate a domain from inside and outside by uniform domains which are extension domains for BV and Sobolev functions.Therefore, we will focus here only on connecting our approach of the more general existence result obtained in Section 3 with other results on the structure of minimizers in PI-spaces.Here with a PI-space we mean a complete metric measure space (X, d, m) where the measure is doubling and the space satisfies a local (1, 1)-Poincaré inequality.Recall that a measure m is doubling on X if there exists a constant C > 0 so that for every x ∈ X and r > 0 we have m(B(x, 2r)) ≤ Cm(B(x, r)).
A metric measure space satisfies a local (1, 1)-Poincaré inequality if there exist constants C > 0 and λ ≥ 1 so that for every function f in X with an upper gradient g f , every x ∈ X and r > 0 we have where f A denotes the average of f in a set A ⊂ X of positive and finite measure.The proof of Theorem 3.4 is based on the minimization of the functional If we replace the term λm(Ω \ A) by λm(Ω∆A) we obtain a more studied functional A minimization of the functional M λ leads to a set which is close in measure to Ω, but not necessarily contained in Ω.Still, the argument in the proof of Theorem 3.4 for showing that the minimizer is a BV -extension set works also for the functional M λ provided that the minimizer has a closed representative (in order to use Lemma 3.2).Since in general we do not know if the minimizer of M λ or M λ has a closed representative, instead of using a global minimizer we took a minimal element in a decreasing chain of closed sets.Recall that by Example 3.7 we know that the minimizer need not have an open representative.
In PI-spaces we do have a closed representative for the global minimizer of M λ in the class of Borel sets.This can be seen via the regularity results of quasiminimal sets.By [2, Proposition 3.20 and Remark 3.23] we have that in PI-spaces the minimizer of the functional M λ is locally K-quasiminimal in X. Recall that a Borel set E ⊂ X is said to be K-quasiminimal, or to have K-quasiminimal boundary in an open set Ω ⊂ X, if for all open U ⋐ Ω and every Borel sets F, G ⋐ U we have A set E is said to be locally K-quasiminimal in Ω, if instead of requiring the minimality for all open U ⋐ Ω we require that for every x ∈ Ω the exists an open neighbourhood V ⊂ Ω of x so that for all U ⋐ V the above holds.
By [16,Theorem 4.2] a K-quasiminimal set in a PI-space has a representative for which the topological and measure theoretic boundaries agree.Recall that the measure theoretic boundary of E consists of those points where the (upper) density of both E and X \ E are positive.By a density point argument, the measure theoretic boundary has always measure zero.Consequently, a K-quasiminimal set has both an open and a closed representative.The proof of Theorem 3.4 then gives that the closed representative is a BV -extension set.However, as we will see in Example 4.1, being a BV -extension set is not invariant under taking representatives, so we cannot conclude directly that the open representative is also a BV -extension set.
Notice also that for the functional M λ we have the local K-quasiminimality only inside Ω.Therefore, via [16,Theorem 4.2] we only know that the topological boundary of the minimizer has measure zero inside Ω.However, if we start with a domain Ω with m(∂Ω) = 0, we can conclude that also the minimizer of M λ has both an open and a closed representative.
The above argumentation leads to natural questions: In a PI-space, is every domain with locally quasiminimal surface a BV -extension set?Is the closure of a domain with locally quasiminimal surface a BV -extension set?We end this section with an example showing that the answer to the first question is negative.In fact, the example shows that even the open representative of a minimizer of M λ need not be a BV -extension set in a PI-space.The same example also answers a question in [16]: domains with locally quasiminimal surface need not be local John domains in PI-spaces.
Recall that a domain Ω is a local John domain if there exist constants C, δ > 0 such that for every x ∈ ∂Ω, every 0 < r < δ and all y ∈ B r (x) ∩ Ω there exists a point z ∈ B Cr (x) ∩ Ω with d(y, z) ≥ r/C and a curve γ ⊂ Ω such that ℓ(γ y,w ) ≤ Cdist(w, ∂Ω) for all w ∈ γ, where γ y,w is the shortest subcurve of γ joining y and w, and ℓ(α) denotes the length of a curve α.A motivation for asking about the local John condition comes from the Euclidean setting, where David and Semmes showed that bounded sets with quasiminimal boundary surfaces are locally John domains [10].
Example 4.1.Consider the metric measure space X = X 1 ∪ X 2 ∪ X 3 where X i = {i} × [0, 1] and for every t ∈ [0, 1] the points (i, t, 0), i = {1, 2, 3} are identified.(Later on we will not always write the first coordinate that was above used only as a label.)Let us write the common part of X i as D = X 1 ∩ X 2 ∩ X 3 .In other words, X = [0, 1] × T , with T being a tripod with unit length legs.The distance d on X is the length distance on each X i given by and the reference measure m is the sum of weighted Lebesgue measures on each X i : The obtained metric measure space (X, d, m) is an Ahlfors 2-regular and satisfies the (1, 1)-Poincaré inequality.We will consider a domain Ω ⊂ X as Ω = Ω 1 ∪ Ω 2 ∪ Ω 3 , where each Ω i ⊂ X i is defined as follows.We start by defining as a basic building block a triangle Now, for the set Ω 1 in X 1 we simply choose and the set Ω 3 ⊂ X 3 is given by The common part J ⊂ D for the sets above is defined by See Figure 1 for an illustration of the domain Ω. 1 lives in three copies of the unit square, X 1 , X 2 , and X 3 , that are glued together at one edge.The domain minimizes M λ and thus has locally quasiminimal surface.One intuitive way to see the quasiminimality is to observe that with local variations one cannot decrease the perimeter much when trying to remove the slits appearing at the common edge in the X 1 ∪ X 2 square.The slits prevent the domain from being locally John or BV -extension domain.
Claim 1: For any λ ≥ 1, the domain Ω is a minimizer of M λ among Borel subsets of X.To prove it, we first show that This can be verified by simply taking as a sequence (f n ) n of Lipschitz functions approaching to χ Ω in L 1 (m) whose elements f n are given by Then, denoting we have that Thus, it remains to estimate the measure of Ω n .Notice that by the choice of the distance d and the slopes in the triangle T , we have for k ∈ {2, 3} that Claim 3: Ω is not locally John domain.To show this, we take as the center x := (0, 0) ∈ ∂Ω.Given any C ≥ 1 and δ > 0 we take k ∈ N large enough so that Notice that by the selection of k we have 0 < r < δ.Then so the point z in the John condition is forced to be selected outside E k .Consequently, any curve γ joining y and z in Ω must pass through a point where in the last inequality we used again the selection of k.This contradicts the John condition with the given parameters C and δ.
Remark 4.2.Notice that as a minimizer of M λ in a PI-space, the domain Ω of Example 4.1 also has quasiminimal surface.If we use as the measure m in the example the 2-dimensional Hausdorff measure, we have that the space (X, d, m) is isotropic.(Let us recall that a metric measure space is isotropic whenever the density function θ E associated with the set of finite perimeter E and for which it holds that Per(E, •) = θ E H | ∂ e E is independent on the set E itself.We refer to [1] for more details about the mentioned density function.)Since the property of being quasiminimal is invariant under a change of the reference measure to a comparable one, we will thus obtain a version of the example where the space is isotropic, but the domain only has quasiminimal surface instead of being a minimizer of M λ .Notice also, that changing to a distance d induced by the Euclidean distances in X i we also preserve the quasiminimality, since the change in distance is bi-Lipschitz.

Open questions
Our extension result leads to several questions that we have not yet been able to answer.In Theorem 3.4 we proved that we can approximate domains from inside by closed BV -extension sets.For the special case of PI-spaces, in Section 4, we noted that minimizers of M λ have also open representatives.However, Example 4.1 showed that the open representatives need not be BV -extension sets even in PI-spaces.What still remained open is if being a minimizer of M λ is really needed or if having just quasiminimal surface is enough: Question 5.1.Let (X, d, m) be a PI-space and Ω ⊂ X a bounded domain with locally K-quasiminimal surface.Is then Ω a BV -extension set?Another question stemming from the proof of Theorem 3.4 is if we really need to take the partial order into use to guarantee that the minimal element has a closed representative.Question 5.2.Let (X, d, m) be a metirc measure space and Ω ⊂ X a bounded domain.Let E be a minimizer of M λ (or M λ ) among Borel subsets of Ω (or X respectively).Does E have a closed representative?
For PI-spaces the answer to Question 5.2 is positive for M λ , see again Section 4. Independent of the minimization approach, the obvious question still remaining is: Question 5.3.Let (X, d, m) be a metric measure space, Ω ⊂ X a bounded domain and ε > 0. Does there exist a BV -extension domain A ⊂ Ω such that m(Ω \ A) < ε? None of our approximations is from outside because we argue that the minimizer is an extension set by comparing the value of the functional to value at a modification of the minimizer where we take away an open subset.
Question 5.4.Let (X, d, m) be a metric measure space, Ω ⊂ X a bounded domain and ε > 0. Does there exist a BV -extension domain (or just a BV -extension set) A ⊃ Ω such that m(A \ Ω) < ε?
In addition to knowing the answer to the above questions, it would be interesting to see if we can also approximate domains by Sobolev W 1,p -extension domains in the absence of the local Poincaré inequality.In particular, the case p = 1 is intimately connected to the BV and perimeter extensions even in general metric measure spaces [8].

Corollary 2 . 5 .
Let (X, d, m) be a metric measure space.Let B ⊂ X be closed and define Y = (B, d| B×B , m| B ). Then BV (B) = BV (Y ) and the total variation measures |Df | B and |Df | Y agree on the Borel subsets of B for every f ∈ BV (B).Moreover,

Step 3 :
Fix any λ > 0 and C ∈ C Ω,λ .Let G λ,C be a minimal element in {A ∈ C Ω,λ : A ≺ λ C} with respect to the partial order ≺ λ .Then we have that E G λ < +∞.Proof of Step 3. By Proposition 3.1, we only need to check that the zero extension is bounded for characteristic functions of open sets of finite perimeter in G λ,C .So, let A ⊂ G λ,C be relatively open with Per G λ,C (A) < +∞.Then by the minimality of G λ,C and the fact that m

3 Figure 1 .
Figure1.The domain Ω in Example 4.1 lives in three copies of the unit square, X 1 , X 2 , and X 3 , that are glued together at one edge.The domain minimizes M λ and thus has locally quasiminimal surface.One intuitive way to see the quasiminimality is to observe that with local variations one cannot decrease the perimeter much when trying to remove the slits appearing at the common edge in the X 1 ∪ X 2 square.The slits prevent the domain from being locally John or BV -extension domain.