Existence of isoperimetric regions in sub-Finsler nilpotent groups

Julián Pozuelo

doi:10.1515/agms-2025-0025

Article Open Access

Existence of isoperimetric regions in sub-Finsler nilpotent groups

Julián Pozuelo

Published/Copyright: July 31, 2025

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From the journal Analysis and Geometry in Metric Spaces Volume 13 Issue 1

Abstract

We consider a nilpotent Lie group with a bracket-generating distribution ℋ and an asymmetric left-invariant norm ∣ ⋅ ∣ K induced by a convex body K ⊆ R k containing 0 in its interior. In this study, we prove the existence of minimizers of the perimeter functional P K associated with ∣ ⋅ ∣ K under a volume (Haar measure) constraint.

Keywords: isoperimetric regions; Nilpotent Lie groups; sub-Riemannian geometry

MSC 2010: 53C17; 49Q20

1 Introduction

A well-known problem in Euclidean space is the problem of minimizing the perimeter associated with a convex body K under a volume constraint. Here, the volume is the Lebesgue measure ∣ ⋅ ∣ and the K -perimeter or Finsler perimeter is given by

P K ( E ) = sup ∫ E div U d x : U ∈ C 0 1 ( R d ; R d ) , ∣ U ∣ K , ∞ ⩽ 1 ,

where ∣ ⋅ ∣ K is an asymmetric norm in R d associated with a convex body K . In case K is the unit disk in R d , the K -perimeter is the Euclidean perimeter. When P K ( E ) < + ∞ , E is said to be a finite K -perimeter set. The isoperimetric profile is the function

I K ( v ) = inf { P K ( F ) : F ⊆ R d finite K -perimeter , ∣ F ∣ = v } .

The sets (if they exist) E ⊆ R d such that

P K ( E ) = I K ( v ) ,

are called Wulff shapes and were described by the crystallographer G. Wulff in 1895. A first mathematical proof of the isoperimetric property of Wulff shapes was given by Dinghas [13]. Other versions of Wulff’s results were given by Busemann [9], Taylor [47], Fonseca [14], and Fonseca and Müller [15] (see also Gardner [20] and Burago and Zalgaller [8]).

In recent years, there has been an increase of interest in the isoperimetric problem in the context of metric measure spaces. The privileged position of Carnot groups within geometric measure theory is revealed by its characterization as the only metric spaces that are locally compact, geodesic, isometrically homogeneous, and self-similar (i.e., admitting a dilation). This characterization can be found as Theorem 1.1 in [28]. Removing the self-similarity condition, sub-Finsler nilpotent groups acquire relevant importance. Pansu [40] proved an isoperimetric inequality in the Heisenberg group and conjectured that some spheres are the isoperimetric regions. Garofalo and Nhieu [21] proved an isoperimetric inequality in R n with a family of vector fields satisfying a Hörmander condition assuming a doubling property, a Poincaré inequality and that R n with the sub-Riemannian distance is a complete length-space. There are a number of partial results on the sub-Riemannian isoperimetric problem in the Heisenberg group that can be found in [35,36,43,45].

For a different notion of perimeter of a submanifold of fixed degree immersed in a graded manifold, see [10] (also [22,32]).

Considering an asymmetric left-invariant norm in the horizontal distribution of the Heisenberg group, we obtain the sub-Finsler Heisenberg group. In this group, the isoperimetric problem for the Minkowski content was considered in [46], while two results on the sub-Riemannian isoperimetric problem in the Heisenberg group were generalized to its sub-Finsler counterpart in [16,42].

There is a variety of existence results of isoperimetric regions in Riemannian geometry, such as Morgan’s result in [37] when a subgroup of the isometry group of M acts cocompactly on M . Different existence results in Riemannian geometry are obtained in [30,34,39,44]. When M is a compact Riemannian manifold, existence follows from the direct method in calculus of variations. Recent existence results have been proven in Riemannian manifolds M assuming that Ricci curvature is nonnegative and Euclidean volume growth of geodesic balls [2], and assuming a lower bound on the Ricci curvature and that M is asymptotic in Gromov-Hausdorff sense to a space form [3].

In sub-Riemannian geometry, apart from the compact case, there are only two known results. Galli and Ritoré proved in [19] an existence result in contact sub-Riemannian manifolds. The argument followed Morgan’s structure: they pick a minimizing sequence of sets of volume v whose perimeters approach the infimum of the perimeters of sets of volume v . This sequence can be split into two subsequences. The first subsequence is converging to a set, and it is proved that it is isoperimetric for its volume and bounded. Nevertheless, there might be a loss of mass at infinity. In this case, they use isometries to translate the second subsequence, which is diverging, to recover some of the lost volume. An essential point is that they always recover a fixed fraction of the volume. This problem has been treated in [4] in metric measure spaces satisfying the Riemannian curvature-dimension condition. In Carnot groups, existence of isoperimetric regions was proven by Leonardi and Rigot [29]. Dilations in a Carnot group G play a key role, since from them it is direct that the isoperimetric profile has the form I G ( v ) = C v q , where C is a positive constant and q ∈ ( 0 , 1 ) . In particular, the function I G is concave, a crucial property to deduce that there is no loss of mass at infinity (see also [2]).

In this article, we shall consider a nilpotent group G with a set of left-invariant vector fields X satisfying Hörmander’s condition, and an asymmetric left-invariant norm ∣ ⋅ ∣ K associated with a convex body K , without the assumption of being equipped with a family of dilations. The main result of this study is the following.

Theorem 1.1

(Existence of isoperimetric regions) Let ( G , X , K ) be a sub-Finsler nilpotent group. Then, for any v > 0 , there exists a finite perimeter set E such that ∣ E ∣ = v and I K ( v ) = P K ( E ) . Moreover, E has a finite number of connected components.

This result is an extension of the existence result of Leonardi and Rigot for nilpotent groups with no dilations and a sub-Finsler norm. The proof follows the arguments in [19]. As in [2,19], one of the main difficulties is to prove a deformation lemma that allows us to increase the volume of any finite perimeter set while modifying the perimeter in a controlled way; more precisely, the difference of the perimeters is linear with respect to the volume we are adding.

Theorem 1.2

(Deformation lemma) Let ( G , X , K ) be a sub-Finsler nilpotent group. Let F ⊆ G be a finite perimeter set with ∣ F ∣ < + ∞ , and suppose that there exists p ∈ int ( F ) . Then, there exist C def > 0 , γ < 0 , and a family of finite perimeter sets { F λ } γ ⩽ λ < + ∞ with ∣ F λ ∣ = ∣ F ∣ + λ and

(1.1) P K ( F λ ) − P K ( F ) ⩽ C def ∣ λ ∣ .

There are a large number of deformation lemmas in the literature; some of them can be seen in [2,5,19,31,34,37,39]. Theorem 1.2 can be proven in a Riemannian manifold by means of the formulas for the first variation of the volume and the area, as can be seen in [31]. Another path to prove (1.1) relies on a calibration argument (see [2,19]) that consists in taking a small deformation of F , F λ , with volume ∣ F ∣ + λ and such that F λ \ F is foliated by smooth hypersurfaces with bounded mean curvature, and apply a suitable divergence theorem to F λ \ F and the vector field given by the normal to the hypersurfaces in the foliation. In this article, we give an elementary proof of Theorem 1.2 that does not require a divergence theorem. We consider a Cheeger set D inside a ball B r ⊂ int ( F ) and move through a one-parameter family of translations. When the translation of D has positive volume outside F , the perimeter added is controlled using the definition of Cheeger set. By left-invariance of perimeter and volume, the constant C def depends only on the radius of a ball inside int ( F ) . A key property in the proof is inequality (3.6), which can be interpreted as Cheeger sets having bounded mean curvature. This property for Euclidean balls can also be seen in [23]. The dependence of C def only on the radius will allow us to prove a uniform deformation Lemma 5.4 for isoperimetric regions and obtain as a consequence that the isoperimetric profile is locally Lipschitz in Corollary 5.5. The locally Lipschitz property of the isoperimetric profile in RCD spaces is a consequence of the concavity of the isoperimetric profile (see Corollary 4.14 in [6]).

By a calibration argument, we obtain in Proposition 4.1 that the isoperimetric profile is non-decreasing. The property of sub-additiveness of the isoperimetric profile is proven in Corollary 5.3. We shall also extend the properties obtained in Carnot groups by Leonardi and Rigot in [29], that isoperimetric regions are bounded, and its topological and essential boundaries coincide. This result has been proven in [7] in any PI space assuming that a deformation lemma is available (see also [5]). Since Theorem 1.2 needs a priori the existence of an interior point, we cannot use directly the results in [7].

This article is organized as follows. In Section 2, we fix notation and give some background on sub-Finsler nilpotent groups and the notion of K -perimeter. In Section 3, we prove Theorem 1.2. In Section 4, we study some properties of the isoperimetric regions such as that they are open up to a null set (Corollary 4.3), bounded (Theorem 4.9), and its essential and topological boundaries coincide (Theorem 4.8), and prove in Proposition 4.1 that the isoperimetric profile is non-decreasing. In Section 5, we prove Theorem 1.1, the existence of isoperimetric regions, the uniform deformation Lemma 5.4, and deduce that the isoperimetric profile is a sub-additive function Corollary 5.3 and locally Lipschitz in Corollary 5.5.

2 Preliminaries

2.1 Notation

Unless otherwise specified, we let k , d ∈ N , k , d ⩾ 1 . Given A ⊂ R d , we denote A ¯ and A c the closure and the complement of A . The Lebesgue measure and the Euclidean norm in R d will be denoted as ∣ ⋅ ∣ and ∣ ⋅ ∣ e u , respectively. Given a Lie group G we denote ℓ x the left-translation by x ∈ G . Its tangent plane at 0 is the Lie algebra g , and we write [ ⋅ , ⋅ ] for the Lie bracket of vector fields.

2.2 Nilpotent groups

We recall some results on nilpotent groups. For a quite complete description of nilpotent Lie groups, the reader is referred to Section 1.13 in [27] (see also [41]).

Let g be a Lie algebra. We define recursively g 0 = g , g i + 1 = [ g , g i ] = span { [ X , Y ] : X ∈ g , Y ∈ g i } . The decreasing series

g = g 0 ⊇ g 1 ⊇ g 2 ⊇ …

is called the lower central series of g . If g r = 0 and g r − 1 ≠ 0 for some r , we say that g is nilpotent of step r . A connected and simply connected Lie group is said to be nilpotent if its Lie algebra is nilpotent. Given a nilpotent group G , then Theorem 1.127 in [27] implies that, after fixing a particular base in g , the exponential map of left-invariant vector fields is a diffeomorphism between R d and G , and the inverse of this map provides coordinates called canonical coordinates of the first kind. The group product can be recovered by the Hausdorff-Campbell-Baker formula (see [27]) and, moreover, the Lebesgue measure of R d coincides with the Haar measure on R d with this product, thanks to Theorem 1.2.10 in [11]. From now on, we shall denote a nilpotent group as ( R d , ⋅ ) . Given a nilpotent group ( R d , ⋅ ) and a system of linearly independent left-invariant vector fields X = { X 1 , … , X k } , we define the distributions

ℋ 0 ≔ span ( X ) ℋ n ≔ ℋ n − 1 + [ ℋ n − 1 , ℋ 0 ] ,

where n ⩾ 1 . We say that X is a bracket-generating system if ℋ s − 1 ( 0 ) = g for some s ⩾ 1 . The distribution ℋ 0 is called the horizontal distribution. A vector field U is said to be horizontal if U ( x ) ∈ ℋ x 0 for all x in R d . We consider V a complementary subbundle of ℋ 0 in g and a left-invariant Riemannian metric g ≔ ⟨ ⋅ , ⋅ ⟩ , making X an orthonormal basis of ℋ x 0 at every point x ∈ R d , and making orthogonal the subbundles ℋ 0 and V . Given a nilpotent group G = ( R d , ⋅ ) and a bracket-generating system X , there is a natural left-invariant distance d X in R d associated with X called the CC-distance [38]. Geodesic balls associated with d X are called the sub-Riemannian balls and the sub-Riemannian ball with center x ∈ R d and radius r > 0 is denoted by B ( x , r ) . We define the dimension at infinity and the local dimension, denoted by D = D ( G ) and l = ( G , X ) , respectively, as

D = ∑ i = 1 r i n i , l = ∑ i = 1 s i m i ,

where n i ≔ dim ( g i − 1 ) − dim ( g i ) ∀ i ⩾ 1 , m i ≔ dim ( ℋ i − 1 ) − dim ( ℋ i − 2 ) ∀ i ⩾ 2 , m 1 ≔ dim ( ℋ 0 ) and the constants r , s ⩾ 1 are defined as mentioned earlier.

Remark 2.1

It is clear that D ⩾ l . The equality D = l holds if and only if ( R d , ⋅ ) is stratifiable, i.e., g admits a direct-sum decomposition g = V 0 ⊕ V 1 … ⊕ V r − 1 such that V r − 1 ≠ { 0 } and [ V 0 , V j ] = V 1 + j , where 0 ⩽ j ⩽ r − 2 and V r = { 0 } , and moreover, V 0 coincides with the distribution ℋ 0 at the origin.

The following result can be seen in Section IV.5 of [49].

Theorem 2.2

(Propositions IV.5.6 and 5.7 in [49]) Let ( R d , ⋅ ) be a nilpotent group with a bracket-generating system X, and let D and l be the dimension at infinity and the local dimension, respectively. Then, there exist positive constants α and β such that

(2.1) α − 1 t l ⩽ ∣ B ( 0 , t ) ∣ ⩽ α t l , 0 ⩽ t ⩽ 1 , β − 1 t D ⩽ ∣ B ( 0 , t ) ∣ ⩽ β t D , t > 1 .

In particular, from the left-invariance of the distance and the volume, it holds the following inequality:

(2.2) ∣ B ( x , s ) ∣ ⩽ α 2 s r l ∣ B ( x , r ) ∣ ,

where x ∈ R d , 0 < r ⩽ s ⩽ 1 .

2.3 Sub-Finsler norms

We follow the approach developed in [42]. We say that ‖ ⋅ ‖ : R k → [ 0 , + ∞ ) is a norm if it verifies

‖ v ‖ = 0 ⇔ v = 0 .
‖ s v ‖ = s ‖ v ‖ ∀ s > 0 and ∀ v ∈ R k .
‖ u + v ‖ ⩽ ‖ u ‖ + ‖ v ‖ ∀ u , v ∈ R k .

We stress the fact that we are not assuming the symmetry property ‖ − v ‖ = ‖ v ‖ . Associated with a given norm ‖ ⋅ ‖ in R k , we have the set F = { u ∈ V : ‖ u ‖ ⩽ 1 } , which is compact, convex, and includes 0 in its interior. Reciprocally, given a compact convex set K with 0 ∈ int ( K ) , the function

∣ u ∣ K = inf { λ ⩾ 0 : u ∈ λ K }

defines a norm in R k so that F = { u ∈ R k : ∣ u ∣ K ⩽ 1 } . Given a norm ‖ ⋅ ‖ and the Euclidean scalar product ⟨ ⋅ , ⋅ ⟩ e u in R k , we consider its dual norm ‖ ⋅ ‖ * of ‖ ⋅ ‖ with respect to ⟨ ⋅ , ⋅ ⟩ e u defined by

(2.3) ‖ u ‖ * = sup ‖ v ‖ ⩽ 1 ⟨ u , v ⟩ e u .

Given u ∈ R k \ { 0 } , then the compactness of the unit ball of ‖ ⋅ ‖ guarantees the existence of a vector v ∈ ∂ K where the supremum of (2.3) is attained, i.e.,

(2.4) ‖ u ‖ * = ⟨ u , v ⟩ e u .

Now, we move to a nilpotent group with a bracket-generating system X of linearly independent vector fields. Given a convex set K ⊆ R k with 0 ∈ int ( K ) , we consider the left-invariant norm ∣ ⋅ ∣ K in ℋ 0 given by

(2.5) ∑ i = 1 k a i X i ( p ) K p = ∣ ( a 1 , … , a k ) ∣ K ,

for any horizontal vector v = ∑ i a i X i ( p ) ∈ ℋ p 0 and any p ∈ R d . Similarly, we denote by ∣ ⋅ ∣ K , ∗ the dual norm of ∣ ⋅ ∣ K and its extension to ℋ 0 and, given x ∈ R d and an horizontal vector u ∈ ℋ x 0 , we denote by Π K ( u ) ⊂ ℋ x 0 the set of horizontal vectors with ∣ v ∣ K = 1 and satisfying (2.3).

We say that ( R d , … , X , K ) is a sub-Finsler nilpotent group.

2.4 Sub-Finsler perimeter

Throughout this subsection, we consider a sub-Finsler nilpotent group ( R d , · , X , K ) with ∣ ⋅ ∣ K the associated norm in ℋ 0 . Let E ⊂ R d be a measurable set, and Ω ⊂ R d an open set. We say that E has finite K -perimeter in Ω if

P K ( E ; Ω ) = sup ∫ E div ( U ) d x : U ∈ ℋ 0 1 ( Ω ) , ∣ U ∣ K , ∞ ⩽ 1 < + ∞ .

In this expression, ℋ 0 1 ( Ω ) is the space of horizontal vector fields of class C 1 with compact support in Ω , ∣ U ∣ K , ∞ = sup p ∈ Ω ∣ U p ∣ K and the divergence of U = ∑ i = 1 k u i X i is defined as

(2.6) div ( U ) = ∑ i = 1 k X i ∗ ( u i ) ,

where X i ∗ is the adjoint operator in L 2 ( R d ) . This definition can be seen in [18] for an arbitrary family of bracket-generating vector fields. From the expression of the product in canonical coordinates (see [41]), it follows that the Lebesgue measure is also invariant by right translations and, as can be seen in Lemma 1.30 of [50], it holds that the vector fields X i are self-adjoint operators in L 2 ( R d ) , and therefore, div ( U ) = ∑ i = 1 k X i ( u i ) . In case that Ω = R d , we write P K ( E ) ≔ P K ( E ; Ω ) . When K is the unit Euclidean ball in R k centered at 0, we recover the sub-Riemannian perimeter (cf. [18])

P ( E ; Ω ) = sup ∫ E div ( U ) d x : U ∈ ℋ 0 1 ( Ω ) , ∣ U ∣ e u , ∞ ⩽ 1 < + ∞ ,

where ∣ U ∣ K , ∞ = sup p ∈ Ω ∣ U p ∣ e u and ∣ ⋅ ∣ e u denotes the Euclidean norm.

It is well known the equivalence between any norm and ∣ ⋅ ∣ e u , i.e., there exists a constant C K ⩾ 1 such that

(2.7) C K − 1 ∣ x ∣ e u ⩽ ∣ x ∣ K ⩽ C K ∣ x ∣ e u , ∀ x ∈ R d .

Let E ⊂ R d be a measurable set and Ω ⊂ R d an open set. Take U ∈ ℋ 0 1 ( Ω ) a vector field with ∣ U ∣ K , ∞ ⩽ 1 . Hence, ∣ C K − 1 U ∣ e u ⩽ ∣ U ∣ K ⩽ 1 and

∫ E div ( U ) d x = C K ∫ E div ( C K − 1 U ) d x ⩽ C K P ( E ; Ω ) .

Taking supremum over the set of C 1 horizontal vector fields with compact support in Ω and ∣ ⋅ ∣ K ⩽ 1 , we obtain P K ( E ; Ω ) ⩽ C K P ( E ; Ω ) . In a similar way, we obtain the inequality C K − 1 P ( E ; Ω ) ⩽ P K ( E ; Ω ) , so that we have

(2.8) C K − 1 P ( E ; Ω ) ⩽ P K ( E ; Ω ) ⩽ C K P ( E ; Ω ) .

As a consequence, E has finite K -perimeter if and only if it has finite (sub-Riemannian) perimeter. Hence, we have the following theorem [21].

Theorem 2.3

Let F n be a sequence of measurable sets with P K ( F n ) ⩽ c for some constant c < + ∞ . Then, there exists a subsequence converging in L loc 1 ( R d ) to a measurable set F with P K ( F ) < + ∞ .

The proof of the Riesz representation theorem can be adapted (see §2.4 in [42]), to obtain the existence of a Radon measure ∣ ∂ E ∣ K on R d and a ∣ ∂ E ∣ K -measurable horizontal vector field ν K in R d with ∣ ν K ∣ K , ∗ = 1 ∣ ∂ E ∣ K -almost everywhere, so that

(2.9) ∫ E div ( U ) d x = ∫ R d ⟨ U , ν K ⟩ d ∣ ∂ E ∣ K ,

where U is a C 0 1 horizontal vector field. We remark that in the particular case K = D , the k -dimensional Euclidean ball of radius 1 centered at 0, then ∣ ∂ E ∣ ( Ω ) and ν D are the sub-Riemannian perimeter of E in Ω and the horizontal unit normal of E , respectively, as defined in Section 2.3 of [18], and shall be denoted as ∣ ∂ E ∣ ( Ω ) = P ( E ; Ω ) and ν , respectively. Moreover, following Section 2.4 in [42], we obtain the representation formula

(2.10) ∣ ∂ E ∣ K = ∣ ν ∣ K , ∗ ∣ ∂ E ∣ .

From the construction of the measure ∣ ∂ E ∣ K , for any open set Ω ⊆ R d , the following equality holds:

P K ( E ; Ω ) = ∣ ∂ E ∣ K ( Ω ) .

The following properties of the perimeter are deduced from the definition and the invariance by left-translations of the Lebesgue measure:

P ( ⋅ , Ω ) is lower semicontinuous with respect to the L loc 1 ( Ω ) convergence.
(Left-invariance) P K ( ℓ x ⋅ E ) = P K ( E ) for all x ∈ R d .
(Locality) P K ( E ; Ω ) = P ( F ; Ω ) whenever ( E △ F ) ∩ Ω has zero measure, where E △ F is the symmetric difference.

Let E be a finite perimeter set and Ω an open set. Using Theorem 3.2.3 in [18] with f ( x , η ) = ∣ η ∣ K , ∗ , we obtain the following representation of the perimeter:

P K ( E , Ω ) = inf liminf n → + ∞ ∫ Ω ∣ ∇ X u n ∣ K , ∗ d x : u n ∈ Lip loc ( Ω ) , u n → χ E in L loc 1 ( Ω ) ,

where χ E is the characteristic function of E , ∇ X u = ( X 1 u , … , X k u ) and Lip loc ( Ω ) is the space of locally Euclidean Lipschitz real-valued functions in Ω . From this representation, it follows (see [33] and [1]) that

P ( F ; Ω ) = P ( F c ; Ω ) .

We remark that this property is not true for all perimeters P K . Moreover, reasoning as in [33], we obtain that for any finite perimeter sets E and F , it holds

(2.11) P K ( E ∩ F ; Ω ) + P K ( E ∪ F ; Ω ) ⩽ P K ( E ; Ω ) + P K ( F ; Ω ) .

We shall use exhaustively the following decomposition of the K -perimeter:

(2.12) P K ( E ) ⩾ P K ( E ∩ B ) + P K ( E \ B ) − 2 C K P ( E ∩ B ; ∂ B ) ,

where B is any sub-Riemannian ball in R d , ∂ B is its topological boundary, and P is the sub-Riemannian perimeter. Indeed, from the locality property, we have

P K ( E ∩ B ) = P K ( E ∩ B ; B ) + P K ( E ∩ B ; ∂ B ) + P K ( E ∩ B ; B ¯ c ) = P K ( E ; B ) + P K ( E ∩ B ; ∂ B ) .

From the aforementioned equation and the relation between the Euclidean and the Minkowski norm (2.8), we obtain

(2.13) P K ( E ; B ) = P K ( E ∩ B ) − P K ( E ∩ B ; ∂ B ) ⩾ P K ( E ∩ B ) − C K P ( E ∩ B ; ∂ B ) .

Similarly, it holds

(2.14) P K ( E ; B ¯ c ) ⩾ P K ( E \ B ) − C K P ( E \ B ; ∂ B ) = P K ( E \ B ) − C K P ( E ∩ B ; ∂ B ) ,

where we used P ( F ; Ω ) = P ( F c ; Ω ) . Adding (2.13) and (2.14), we obtain (2.12).

The following relation between the sub-Riemannian perimeter and the derivative of the volume can be found as Lemma 3.5 in [1].

Lemma 2.4

Let ( R d , · , X , K ) be a sub-Finsler nilpotent group. Let F ⊆ R d be a finite (sub-Riemannian) perimeter set and B r the sub-Riemannian ball of radius r centered in 0. Then, for a.e. r > 0 , we have

(2.15) max { P ( F ∩ B r ; ∂ B r ) , P ( F \ B r ; ∂ B r ) } ⩽ d d s s = r m ( s ) ,

where m ( s ) = ∣ E ∩ B ( x , s ) ∣ .

A measurable set F ⊆ R d is said to have density s ∈ [ 0 , 1 ] at a point x ∈ R d , provided the following limit exists and equals s

lim r → 0 + ∣ F ∩ B ( x , r ) ∣ ∣ B ( x , r ) ∣ .

The set of points where the density of F is s is denoted by F s . The essential boundary of F is R d \ ( F 1 ∪ F 0 ) .

2.5 Isoperimetric inequality for small volumes

A fundamental tool in a metric measure space ( X , d , μ ) is the existence of a (1, 1)-Poincaré inequality, i.e., the existence of constants C ⩾ 0 and λ ⩾ 1 such that

(2.16) ∫ B ( x , r ) ∣ f − f x , r ∣ d μ ⩽ C r ∫ B ( x , λ r ) ∣ ∇ f ∣ d μ ,

for all f locally Lipschitz, where B ( x , r ) is the metric ball of center x and radius r , f x , r ≔ 1 ⁄ ∣ B ( x , r ) ∣ ∫ B ( x , r ) f d μ and ∣ ∇ f ∣ is an upper gradient of f in the sense of Heinonen and Koskela [25]. In the context of connected Lie groups with polynomial volume growth, a (1, 1)-Poincaré inequality was proven by Varopoulos in [48], and in R d with a Lie bracket-generating system by Jerison in [26]. As stated by Hajłasz and Koskela in Theorem 5.1 and 9.7 in [24], the (1, 1)-Poincaré inequality (2.16) together with (2.2) implies the following Sobolev inequality:

(2.17) ∫ B ( x , r ) ∣ f − f x , r ∣ l ⁄ ( l − 1 ) d μ ( l − 1 ) ⁄ l ⩽ C ˜ r ∫ B ( x , r ) ∣ ∇ f ∣ d μ ,

for r < 1 .

The following isoperimetric inequality for small volumes follows from Proposition 3.19 in [5] by (2.2) and (2.17).

Theorem 2.5

(Isoperimetric inequality for small volumes) Let ( R d , … , X , K ) be a sub-Finsler nilpotent group with local dimension l. There exist C isop > 0 and v 0 > 0 such that if F ⊆ R d is any finite perimeter set and ∣ F ∣ < v 0 , then

(2.18) C isop ∣ F ∣ ( l − 1 ) ⁄ l ⩽ P K ( F ) .

3 Deformation lemma

In this section, we prove Theorem 1.2. We need the existence result of Cheeger sets given in Theorem 3.6 of [17].

Given an open measurable set Ω ⊂ R d , the Cheeger constant of Ω for the perimeter P K is defined as

h Ω = inf P K ( F ) ∣ F ∣ : F ⊆ Ω , ∣ F ∣ > 0 .

A finite perimeter set D ⊆ Ω is called a Cheeger set of Ω for the perimeter P K if h Ω = P K ( D ) ⁄ ∣ D ∣ . We say that a finite perimeter set D is self-Cheeger if it is a Cheeger set of itself. Note that any Cheeger set of Ω is self-Cheeger.

Theorem 3.1

Let ( X , A , m ) be a measure space, Ω ∈ A be an open set with 0 < m ( Ω ) < v 0 and such that there exists F ⊂ Ω with finite perimeter set. Let P : A → R 0 + be a perimeter functional satisfying the following properties:

The perimeter P is lower semicontinuous with respect to the L 1 ( X , m ) convergence.
The family
{ χ F : F ∈ A , F ⊆ Ω , P ( F ) ⩽ c }
is compact in L 1 ( X , m ) for all c ⩾ 0 .
There exists a map f : [ 0 , v 0 ) → R + such that, given 0 < v < v 0 and F ∈ A with m ( F ) ⩽ v , it holds
P ( F ) ⩾ f ( v ) m ( F )
and
lim v → 0 + f ( v ) = + ∞ .

Then, there exists a Cheeger set D of Ω .

Remark 3.2

In the statement of Theorem 3.6 in [17], the function f in property (iii) is defined in [ 0 , + ∞ ) and the inequality holds for any m ( F ) < + ∞ . Nevertheless, it follows from the proof that it is enough that f and the inequality are defined in [ 0 , v 0 ) and that m ( Ω ) < v 0 .

As a significant example, we consider a sub-Finsler nilpotent group ( R d , ⋅ , X , K ) with the perimeters P K and P ˜ K , given by

P ˜ K ( F ) = P K ( F c ) ,

for any finite perimeter set F ⊆ H n . The perimeters P K and P ˜ K are lower semicontinuous with respect to the L loc 1 convergence. Moreover, conditions (ii) and (iii) hold for P K by Theorems 2.3 and 2.5, respectively, where v 0 > 0 is given in Theorem 2.5. Thanks to (2.8) and that the sub-Riemannian perimeter satisfies

(3.1) P ( F ) = P ( F c ) ,

P ˜ K also satisfies conditions (ii) and (iii) by Theorems 2.3 and 2.5, respectively.

Corollary 3.3

Let ( R d , · , X , K ) be a sub-Finsler nilpotent group and F ⊂ R d a finite perimeter set with p ∈ int ( F ) . Then, there exist D and D ˜ self-Cheeger sets for P K and P ˜ K contained in int ( F ) .

Proof

Let p ∈ int ( F ) and let r ′ > 0 such that B ( p , r ′ ) ⊆ int ( F ) and ∣ B ( p , r ′ ) ∣ < v 0 , where v 0 is defined in Theorem 2.5 and B ( p , r ′ ) the sub-Riemannian ball centered at p with radius r ′ > 0 . Since r ↦ ∣ F ∩ B ( p , r ) ∣ is a non-decreasing function, it is differentiable a.e. and, by Lemma 2.4, we have

P ( B ( p , r ) ) = P ( F ∩ B ( p , r ) ) = P ( F ∩ B ( p , r ) ; ∂ B ( p , r ) ) < + ∞ ,

for a.e. r ′ > r > 0 . Using (2.8) and (3.1), it holds that P K ( B ( p , r ) ) and P ˜ K ( B ( p , r ) ) are also finite. Hence, we can apply Theorem 3.1 to B ( p , r ′ ) and the result follows.□

Proof of Theorem 1.2

Let D ⊆ int ( F ) be a self-Cheeger set for P K , and consider the family of sets D t = ℓ ( t , 0 , … , 0 ) D and V t = F ∩ D t where t ⩾ 0 . By left-invariance of the perimeter and the volume, D t is also a self-Cheeger set with the same Cheeger constant, denoted by h . Clearly, ∣ D ∣ − ∣ V 0 ∣ = 0 and since ∣ F ∣ is finite, lim t → + ∞ ∣ V t ∣ = 0 . To check the continuity of ∣ V t ∣ , we fix t 0 and t n → t 0 . We have that

(3.2) ∣ ∣ V t n ∣ − ∣ V t 0 ∣ ∣ = ∫ R d χ V t n ( x ) − χ V t 0 ( x ) d x ⩽ ∫ R d χ F ( x ) ∣ χ D t n ( x ) − χ D t 0 ( x ) ∣ d x ⩽ ∫ R d ∣ χ D t n ( x ) − χ D t 0 ( x ) ∣ d x .

Note that in (3.2), we are denoting by ∣ ⋅ ∣ the Lebesgue measure and the absolute value. Let us check that

(3.3) ∫ R d ∣ χ A t n ( x ) − χ A t 0 ( x ) ∣ d x → 0 ,

when A is a bounded set with smooth boundary and A t n = ℓ ( t n , 0 … , 0 ) A . If p ∈ int ( A t 0 ) ∪ ext ( A t 0 ) , it easily follows that χ A t n ( p ) → χ A t 0 ( p ) . Hence, χ A t n → χ A t 0 a.e. since ∂ A t 0 has measure 0. Moreover, since A t 0 ∪ { A t n : n ∈ N } is bounded, we can apply the Lebesgue-dominated convergence to ∣ χ A t n − χ A t 0 ∣ and (3.3) holds. Given ε > 0 , by Theorem 2.2.2 in [18], we can take A a bounded set with smooth boundary such that

∫ R d ∣ χ A ( x ) − χ D ( x ) ∣ d x ⩽ ε .

Then, writing A t n = ℓ ( t n , 0 … , 0 ) A and using the invariance of the Lebesgue measure by left-translations and (3.3), we have

(3.4) ∫ R d ∣ χ D t n ( x ) − χ D t 0 ( x ) ∣ d x ⩽ ∫ R d ∣ χ D t n ( x ) − χ A t n ( x ) ∣ d x + ∫ R d ∣ χ A t n ( x ) − χ A t 0 ( x ) ∣ d x + ∫ R d ∣ χ A t 0 ( x ) − χ D t 0 ( x ) ∣ d x ⩽ 2 ε + ∫ R d ∣ χ A t n ( x ) − χ A t 0 ( x ) ∣ d x .

From (3.3), taking limits when t n → t 0 in (3.4) and using that ε > 0 is arbitrary, we obtain that the right-hand side of (3.2) tends to 0 and hence the continuity of ∣ V t ∣ . Therefore, we can take a nonempty open set I , where the map t ↦ ∣ D ∣ − ∣ V t ∣ goes to ( 0 , ∣ D ∣ ) . Moreover,

(3.5) ∣ F ∪ D t ∣ = ∣ F ∣ + ( ∣ D t ∣ − ∣ V t ∣ ) .

Since D t is self-Cheeger, we have the equivalent expressions

(3.6) P K ( D t ) ∣ D t ∣ ⩽ P K ( V t ) ∣ V t ∣ P K ( D t ) ∣ D t ∣ ⩽ P K ( V t ) ∣ D t ∣ − ( ∣ D t ∣ − ∣ V t ∣ ) P K ( D t ) − h ( ∣ D t ∣ − ∣ V t ∣ ) ⩽ P K ( V t ) P K ( D t ) ⩽ P K ( V t ) + h ( ∣ D t ∣ − ∣ V t ∣ ) .

Using (2.11) and (3.6), we obtain

(3.7) P K ( F ∪ D t ) + P K ( F ∩ D t ) ⩽ P K ( F ) + P K ( D t ) ⩽ P K ( F ) + h ( ∣ D t ∣ − ∣ V t ∣ ) + P K ( F ∩ D t ) .

From (3.5) and (3.7), we obtain (1.1) for any 0 ⩽ λ < ∣ D ∣ . Applying the previous reasoning again to F ∪ D t 0 , where ∣ D t 0 ∣ − ∣ V t 0 ∣ = ∣ D ∣ ⁄ 2 , we obtain (1.1) for any 0 ⩽ λ < 3 ∣ D ∣ ⁄ 2 . By induction, we obtain (1.1) for 0 ⩽ λ < + ∞ .

To substract volume, we consider D ˜ ⊆ int ( F ) self-Cheeger set for P ˜ K and define the sets D ˜ t = ℓ ( t , 0 , … , 0 ) D ˜ and V ˜ t = F c ∩ D ˜ t , where t ⩾ 0 . Note that ∣ V ˜ 0 ∣ = 0 and lim t → + ∞ ∣ V ˜ t ∣ = ∣ D ˜ ∣ . Let I ˜ be a nonempty open set such that the map t ↦ ∣ D ˜ ∣ − ∣ V ˜ t ∣ goes to ( 0 , ∣ D ˜ ∣ ) . We have

(3.8) ∣ F \ D ˜ t ∣ = ∣ F ∣ − ( ∣ D ˜ t ∣ − ∣ V ˜ t ∣ )

and

(3.9) P ˜ K ( D ˜ t ) ⩽ P ˜ K ( V ˜ t ) + h ˜ r ( ∣ D ˜ t ∣ − ∣ V ˜ t ∣ ) .

Using Morgan’s Laws, (2.11), and (3.9), we obtain

(3.10) P ˜ K ( F c ∩ D ˜ t ) + P ˜ K ( F c ∪ D ˜ t ) = P K ( F ∪ D ˜ t c ) + P K ( F ∩ D ˜ t c ) ⩽ P K ( F ) + P K ( D ˜ t c ) = P K ( F ) + P ˜ K ( D ˜ t ) ⩽ P K ( F ) + P ˜ K ( F c ∩ D ˜ t ) + h ˜ r ( ∣ D ˜ t ∣ − ∣ V ˜ t ∣ ) .

Moreover, we have

(3.11) P K ( F \ D ˜ t ) = P K ( F ∩ D ˜ t c ) = P K ( ( F c ∪ D ˜ t ) c ) = P ˜ K ( F c ∪ D ˜ t ) .

Substituting (3.11) into (3.10) and using (3.8), we obtain (1.1) for − ∣ D ˜ ∣ < λ ⩽ 0 .□

Remark 3.4

The constant C def depends only on the radius of the ball B ( x , r ) ⊂ int ( F ) . In particular, given F 1 and F 2 two finite perimeter sets with finite volume and with B ( x 1 , r ) ⊆ F 1 and B ( x 2 , r ) ⊆ F 2 , then we can take the same constant C def > 0 in (1.1).

Remark 3.5

Given an open bounded finite perimeter set F with finite volume and a Cheeger set D of F for P K , we can fix C def = h F the constant in (1.1) for positive values of λ . We do not know whether this constant is optimal.

4 Properties of isoperimetric regions

Throughout this section, ( R d , · , X , K ) denotes a nilpotent group with X a bracket-generating system with k linearly independent vector fields, K denotes a convex body in R k with 0 ∈ int ( K ) , and B ( x , r ) is the sub-Riemannian ball centered in x of radius r > 0 . We shall see that an isoperimetric region E is open up to a null set and its topological and essential boundary coincide, using the arguments developed in [12,29]. We shall also prove that isoperimetric regions are bounded.

The isoperimetric profile is defined as

I K ( v ) ≔ inf { P K ( E ) : E ⊆ R d is a finite perimeter set and ∣ E ∣ = v } .

Proposition 4.1

Let ( R d , ⋅ , X , K ) be a sub-Finsler nilpotent group. The isoperimetric profile is non-decreasing.

Proof

Since g \ g 1 ⊆ ℋ 0 0 , we can take without loss of generality a vector field X 1 ∈ X with X 1 ( 0 ) ∉ g 1 , and let D be the distribution orthogonal to X 1 . Clearly, D is integrable, and by Froebenius’ theorem, there exists an orientable hypersurface S passing through 0 with T S = D . Let S + and S − be the open regions in R d with boundary S and horizontal normal vectors X 1 and − X 1 , respectively.

Fix v > w > 0 , and let E n ⊆ R d such that ∣ E n ∣ = v and P K ( E n ) = I K ( v ) + 1 n . Let p n be such that ∣ ℓ p n E n ∩ S + ∣ = w . By abuse of notation, we will write E n and E n − for ℓ p n E n and ℓ p n E n ∩ S − , respectively. Let U ∈ Π K ( X 1 ) , where Π K ( X 1 ) is defined in Section 2.3. Applying (2.9) to E n − and U and K = D the Euclidean unit ball centered at 0, we obtain

(4.1) ∫ E n − div U d x = − ∫ E n ∩ S ⟨ U , X 1 ⟩ d ∣ ∂ E n − ∣ + ∫ S − ⟨ U , ν ⟩ d ∣ ∂ E n − ∣ ,

where ∣ ν ∣ = 1 ∣ ∂ E n − ∣ -almost everywhere. From (2.4) and (2.10), we obtain

(4.2) ∫ S − ⟨ U , ν ⟩ d ∣ ∂ E n − ∣ ⩽ ∫ S − ∣ ν ∣ K , ∗ d ∣ ∂ E n − ∣ = ∣ ∂ E n − ∣ K ( S − ) = P K ( E n − ; S − ) = P K ( E n ; S − ) .

Since U has constant coordinates in X , div U = 0 . Therefore, from (4.1) and (4.2), we obtain

(4.3) P K ( E n ∩ S + ; S ) = ∫ E n ∩ S ∣ ν ∣ K , ∗ d ∣ ∂ E n − ∣ = ∫ E n ∩ S ⟨ U , X 1 ⟩ d ∣ ∂ E n − ∣ ⩽ P K ( E n ; S − ) .

Adding P K ( E n ; S + ) to both sides of (4.3), we obtain

□ I K ( w ) ⩽ P K ( E n ∩ S + ) ⩽ P K ( E n ) = I K ( v ) + 1 n .

We shall need the following lemma proven in [29] for Carnot groups.

Lemma 4.2

Let E be an isoperimetric region, x ∈ R d and 0 < r ⩽ 1 . Then, there exists ε > 0 such that if r − l ∣ B ( x , r ) \ E ∣ ⩽ ε , then

∣ B ( x , r ⁄ 2 ) \ E ∣ = 0 ,

where is B ( x , r ) the sub-Riemannian ball centered in x of radius r > 0 .

Proof

Let ∣ E ∣ = v . Suppose that r − l ∣ B ( x , r ) \ E ∣ ⩽ ε . Let t > 0 , B ≔ B ( x , t ) , E t ≔ E ∪ B , and m ( t ) ≔ ∣ B \ E ∣ . We first check that

(4.4) P K ( E ) ⩾ P K ( E t ) + C K − 2 P K ( B \ E ) − C K P ( E t ; ∂ B ) − C K − 1 P ( B \ E ; ∂ B ) .

It is clear that

P K ( E ) ⩾ P K ( E ; B ) + P K ( E ; B ¯ c ) = P K ( E ; B ) + P K ( E t ; B ¯ c ) = P K ( E ; B ) + P K ( E t ) − P K ( E t ; ∂ B ) ⩾ P K ( E ; B ) + P K ( E t ) − C K P ( E t ; ∂ B ) ,

where we used that P K ( E t ; B ) = 0 . Moreover, we have

P K ( E ; B ) ⩾ C K − 1 P ( E ; B ) = C K − 1 ( P ( B \ E ) − P ( B \ E ; ∂ B ) ) ⩾ C K − 1 ( C K − 1 P K ( B \ E ) − P ( B \ E ; ∂ B ) ) ,

where we used P ( E ; B ) = P ( B \ E ; B ) . From the aforementioned inequalities, (4.4) follows.

Since ∣ E t ∣ ⩾ ∣ E ∣ and E is an isoperimetric region, Proposition 4.1 gives us

(4.5) P K ( E ) ⩽ P K ( E t ) .

Moreover, applying Lemma 2.4 to E c , we obtain

(4.6) max { P ( E t ; ∂ B ) , P ( B \ E ; ∂ B ) } ⩽ m ′ ( t ) ,

where we used B \ E = E c ∩ B and ( E ∪ B ) c = E c \ B . Substituting (4.6) and (4.5) into (4.4) and using the isoperimetric inequality (2.18) to the term P K ( B \ E ) , then for a.e. t > 0 , it follows that

(4.7) C m ( t ) ( l − 1 ) ⁄ l ⩽ m ′ ( t ) .

Suppose that m ( t ) > 0 for all t ∈ [ r ⁄ 2 , r ] ; otherwise, there is nothing to prove. Then, we can rewrite (4.7) as

C ⩽ m ′ ( t ) m ( t ) ( l − 1 ) ⁄ l ,

and integrating between r ⁄ 2 and r ,

r ⩽ C ( m ( r ) 1 ⁄ l − m ( r ⁄ 2 ) 1 ⁄ l ) ⩽ C m ( r ) 1 ⁄ l ⩽ C ε 1 ⁄ l r .

This is impossible for ε small enough, and we obtain a contradiction. Therefore, ∣ B ( x , r ⁄ 2 ) \ E ∣ = 0 .□

Corollary 4.3

Let E be an isoperimetric region. Then, E coincides almost everywhere with the set

E ˜ = { x ∈ R d : ∃ r > 0 s.t. ∣ B ( x , r ) \ E ∣ = 0 } .

Proof

Clearly, E ˜ ⊆ E 1 . Let r > 0 and x ∈ E 1 . By (2.1), we obtain

∣ B ( x , r ) \ E ∣ ∣ B ( x , r ) ∣ ⩾ C ˜ ∣ B ( x , r ) \ E ∣ r l ⩾ 0 ,

for r small enough. Since the left-hand side of the aforementioned inequality converges to 0 as r → 0 , we can apply Lemma 4.2 to obtain x ∈ E ˜ . The result holds since E 1 coincides almost everywhere with E .□

Remark 4.4

In the proof of Corollary 4.3, it is important to have the same exponent l in (2.1) and in the isoperimetric inequality (2.18).

Remark 4.5

From now on, we assume that an isoperimetric region E is exactly E ˜ , and therefore an open set.

Following again the arguments in [29], we use the deformation Lemma 1.2 to prove Lemma 4.6.

Lemma 4.6

Let E be an isoperimetric region, x ∈ R d and 0 < r ⩽ 1 . Then, there exists ε > 0 such that if r − l ∣ E ∩ B ( x , r ) ∣ ⩽ ε , then

∣ E ∩ B ( x , r ⁄ 2 ) ∣ = 0 ,

where is B ( x , r ) the sub-Riemannian ball centered in x of radius r > 0 .

Proof

Let t > 0 , B ≔ B ( x , t ) , and m ( t ) = ∣ E ∩ B ∣ . Thanks to Remark 4.5, E is open and we can take E t ≔ ( E \ B ) m ( t ) as the set given by the deformation Lemma 1.2 with volume ∣ E t ∣ = ∣ E ∣ . Thus,

P K ( E ) ⩽ P K ( E t ) .

On the other hand, using (1.1) and (2.12), we obtain

P K ( E t ) ⩽ P K ( E \ B ) + C def m ( t ) ⩽ P K ( E ) − P K ( E ∩ B ) + 2 C K m ′ ( t ) + C def m ( t ) ⩽ P K ( E ) − C isop m ( t ) ( l − 1 ) ⁄ l + 2 C K m ′ ( t ) + C def m ( t ) .

The aforementioned inequalities give us

C isop m ( t ) ( l − 1 ) ⁄ l − C def m ( t ) ⩽ 2 C K m ′ ( t ) .

For m ( t ) small enough, there exists C > 0 such that

C m ( t ) ( l − 1 ) ⁄ l ⩽ C isop m ( t ) ( l − 1 ) ⁄ l − C def m ( t ) ,

and

C ′ m ( t ) ( l − 1 ) ⁄ l ⩽ m ′ ( t ) .

Again, supposing that m ( t ) > 0 for a.e. t ∈ [ r ⁄ 2 , r ] , we have

C ′ ⩽ m ′ ( t ) m ( t ) ( l − 1 ) ⁄ l ,

and integrating over r ⁄ 2 and r , we obtain a contradiction for ε > 0 small enough.□

Corollary 4.7

Let E be an isoperimetric region. Then, E c coincides almost everywhere with the set

E ˆ = { x ∈ R d : ∃ r > 0 s. t. ∣ E ∩ B ( x , r ) ∣ = 0 } .

Theorem 4.8

Let ( R d , ⋅ , X , K ) be a sub-Finsler nilpotent group, and let E be an isoperimetric region. Then, the topological and essential boundaries of E coincide.

Proof

Let

S = { x ∈ R d : r − l min { ∣ E ∩ B ( x , r ) ∣ , ∣ B ( x , r ) \ E ∣ } > ε ∀ r ⩽ 1 } .

By Lemmas 4.2 and 4.6, the sets E ˜ , E ˆ , and S form a partition of R d . Since E ˜ and E ˆ are open and disjoint, ∂ E ˜ ∪ ∂ E ˆ ⊆ S . On the other hand, S ⊆ ∂ E = ∂ E c . The result holds since E ˜ and E ˆ are equivalent to E and E c .□

Theorem 4.9

(Boundedness) Any isoperimetric region in a sub-Finsler nilpotent group ( R d , · , X , K ) is bounded.

Proof

Let E be an isoperimetric set of volume v , B the sub-Riemannian ball centered in 0 of radius r > 0 , m ( r ) = ∣ E \ B ∣ , and ( E ∩ B ) m ( r ) be the set given by the deformation Lemma 1.2 with ∣ ( E ∩ B ) m ( r ) ∣ = v . Then, we have

(4.8) P K ( E ) ⩽ P K ( ( E ∩ B ) m ( r ) ) .

Using (1.1), (2.15), (2.12), and isoperimetric inequality (2.18), we obtain

(4.9) P K ( ( E ∩ B ) m ( r ) ) ⩽ P K ( E ∩ B ) + C def m ( r ) ⩽ P K ( E ) − P K ( E \ B ) + 2 C K P ( E ∩ B ; ∂ B ) + C def m ( r ) ⩽ P K ( E ) − P K ( E \ B ) − 2 C K m ′ ( r ) + C def m ( r ) ⩽ P K ( E ) − C isop m ( r ) ( l − 1 ) ⁄ l − 2 C K m ′ ( r ) + C def m ( r ) .

From (4.8) and (4.9), we obtain

− 2 C K m ′ ( r ) ⩾ C isop m ( r ) ( l − 1 ) ⁄ l − C def m ( r ) .

As m ( r ) → 0 as r → ∞ , for r big enough, there exists C > 0 such that

C isop m ( r ) ( l − 1 ) ⁄ l − C def m ( r ) ⩾ C m ( r ) ( l − 1 ) ⁄ l .

Let r > 1 and suppose that m ( r ) > 0 . Then,

(4.10) − m ′ ( r ) m ( r ) ( l − 1 ) ⁄ l ⩾ C ,

since

(4.11) ∫ 1 r − m ′ ( s ) m ( s ) ( l − 1 ) ⁄ l = − ∫ m ( 1 ) m ( r ) 1 s ( l − 1 ) ⁄ l d s = m ( 1 ) 1 ⁄ l − m ( r ) 1 ⁄ l .

Integrating (4.10) between 1 and r and using (4.11), we obtain

m ( 1 ) 1 ⁄ l ⩾ C r + m ( r ) 1 ⁄ l ,

and r is bounded.□

We recall the following property for further reference.

Proposition 4.10

There exists a minimizing sequence { E n } with E n bounded sets with volume v and P K ( E n ) ⩽ I K ( v ) + 1 ⁄ n .

Proof

It is enough to show that for any finite perimeter set F of volume v and any ε > 0 , there is a bounded finite perimeter set E with ∣ E ∣ = v and P K ( E ) ⩽ P K ( F ) + ε . Thanks to (2.1), we can take r ′ > 0 small enough such that

∣ B r ′ ∣ < ε r ′ 4 C K ,

where B r ′ is the sub-Riemannian ball centered at 0 with radius r ′ . Moreover, we can take 0 < r < r ′ such that P K ( B r \ F ; ∂ B r ) < ε 4 , since otherwise by (2.15) and (2.8)

∣ B r ′ ∣ ⩾ ∣ B r ′ ∩ F c ∣ = ∫ 0 r ′ d d s s = r ∣ F c ∩ B s ∣ d r ⩾ ∫ 0 r ′ P ( F c ∩ B r ; ∂ B r ) d r = ∫ 0 r ′ P ( B r \ F ; ∂ B r ) d r ⩾ ∫ 0 r ′ C K − 1 P K ( B r \ F ; ∂ B r ) d r ⩾ ∫ 0 r ′ ε 4 C K d r = ε r ′ 4 C K .

It is clear that for R ′ > r big enough, it holds

0 < ∣ F \ B R ′ ∣ < ε 2 C def ,

where C def is obtained from applying the deformation lemma to F ∪ B r . Moreover, we can take R > R ′ so that P K ( F ∩ B R ; ∂ B R ) < ε 4 , since otherwise, by (2.15) and (2.8) we have

∣ F \ B R ′ ∣ = ∫ R ′ + ∞ d d s s = r ∣ F ∩ B s ∣ d r ⩾ ∫ R ′ + ∞ P ( F ∩ B r ; ∂ B r ) d r ⩾ ∫ R ′ + ∞ ε 4 C K .

Note that E = F ∩ B R ∪ B r has ∣ E ∣ = v − ∣ F \ B R ∣ + ∣ B r \ F ∣ = v − M + m and

P K ( E ) ⩽ P K ( F ) + P K ( F ∩ B R ; ∂ B R ) + P K ( B r \ F ; ∂ B r ) ⩽ P K ( F ) + ε 2 .

If λ = M − m > 0 , we consider E λ the set given by Theorem 1.2, which satisfies ∣ E λ ∣ = v . By Remark 3.4, we can take the same constant C def in (1.1) for E and F ∪ B r . Then, by (1.1), we obtain

P K ( E λ ) ⩽ P K ( E ) + C def λ ⩽ P K ( F ) + ε 2 + C def M ⩽ P K ( F ) + ε .

If m − M > 0 , we reason as in the proof of Proposition 4.1 and take E ∩ S + , where S + is the open region whose boundary S has horizontal normal vector X 1 and such that ∣ E ∩ S + ∣ = v . Then, arguing identically as in the proof of Proposition 4.1, we conclude P K ( E ∩ S + ) ⩽ P K ( E ) .□

5 Existence of isoperimetric regions

Throughout this section, K shall denote a convex body in ℋ 0 0 containing 0 in its interior and B ( x , r ) the sub-Riemannian ball centered in x of radius r > 0 . We shall follow the arguments of Galli and Ritoré [19].

The following lemma can be found in [29] for Carnot groups, and in the context of sub-Finsler nilpotent groups, the proof can be done mutatis mutandis.

Lemma 5.1

(Concentration lemma) Let F be a set with finite perimeter and volume. Suppose that there exists m ∈ ( 0 , ∣ B ( 0 , 1 ) ∣ ⁄ 2 ) such that ∣ F ∩ B ( x , 1 ) ∣ < m for all x ∈ R d . Then, there exists C > 0 depending only on l such that

C ∣ F ∣ l P K ( F ) − l ⩽ m .

The following lemma can be found in [30].

Lemma 5.2

Let { E n } be a sequence of uniformly bounded perimeter sets of volumes { v n } converging to v > 0 . Let E be the limit in L loc 1 ( R d ) of E n . Then, there exists a divergence sequence of radii { r n } such that, setting F n = E n \ B ( 0 , r n ) and up to a subsequence, it is satisfied

(5.1) ∣ E ∣ + liminf n → ∞ ∣ F n ∣ = v , P K ( E ) + liminf n → ∞ P K ( F n ) ⩽ liminf n → ∞ P K ( E n ) .

Proof

Take { s n } increasing with s n − s n + 1 > n . We claim that there exists r n in [ s n , s n + 1 ] such that

P ( E n ∩ B ( 0 , r n ) ; ∂ B ( 0 , r n ) ) < v n ⁄ n ,

where P and B r are the sub-Riemannian perimeter and ball of center 0 and radius r , respectively. Otherwise, by (2.15), we have

v n < ∫ s n s n + 1 P ( E n ∩ B t ; ∂ B t ) d t ⩽ ∫ s n s n + 1 d d s s = t ∣ E n ∩ B s ∣ d t ⩽ v n .

Therefore, by (2.12), we obtain

(5.2) P K ( E n ) ⩾ P K ( E n ∩ B r n ) + P K ( E n \ B r n ) − 2 C K P ( E n ∩ B r n ; ∂ B r n ) ⩾ P K ( E n ∩ B r n ) + P K ( F n ) − 2 C K v n ⁄ n .

On the other hand,

(5.3) ∣ E n ∣ = ∣ E n ∩ B r n ∣ + ∣ E n \ B r n ∣ .

Taking inferior limits in n in (5.2) and (5.3), and using the lower semicontinuity, we have the result.□

Proof of Theorem 1.1

Let { E n } k ∈ N be a minimizing sequence of sets with ∣ E n ∣ = v and P K ( E n ) ⩽ I K ( v ) + 1 n . By compactness, the sequence converges in L loc 1 ( R d ) to a set E 0 . Let v 0 = ∣ E 0 ∣ . By Lemma 5.2, we can find a sequence of divergence radii r n such that, writing F n = E n \ B ( 0 , r n ) , we have

(5.4) v 0 + liminf n → ∞ ∣ F n ∣ = v , P K ( E 0 ) + liminf n → ∞ P K ( F n ) ⩽ I K ( v ) .

If v 0 = v , then the theorem is proven. If v 0 < v , we claim that E 0 is isoperimetric for its volume. Otherwise, we use Proposition 4.10 to find O ⊂ G bounded such that ∣ O ∣ = v 0 and P K ( O ) < P K ( E 0 ) . By definition of F n , we can find n 0 such that ∀ n > n 0 , O and F n are disjoint. Then,

liminf n → ∞ ∣ O ∪ F n ∣ = ∣ O ∣ + liminf n → ∞ ∣ F n ∣ = v .

By (5.4),

I K ( v ) ⩽ liminf n → ∞ P K ( O ∪ F n ) = P K ( O ) + liminf n → ∞ P K ( F n ) < P K ( E 0 ) + liminf n → ∞ P K ( F n ) ⩽ I K ( v ) ,

and we have a contradiction.

If v 0 < v , then there exists m > 0 and a divergent sequence of points { x n } such that ∣ F n ∩ B ( x n , 1 ) ∣ ⩾ m . Otherwise, by Lemma 5.1, for any integer k > 0 ,

C 1 ⁄ l ∣ F n ∣ ⩽ 1 k P K ( F n ) ,

which contradicts v 0 < v by (5.4). Hence, the sets ℓ − x n F n converge in L loc 1 ( R d ) to a set E 1 of volume 0 < v 1 ⩽ lim n → ∞ ∣ F n ∣ = v − v 0 . By Lemma 5.2, we can find a divergent sequence { r ′ n } of radii so that the sets F ′ n = ( ℓ − x n F n ) \ B ( 0 , r ′ n ) verifies

(5.5) v 1 + liminf n → ∞ ∣ F ′ n ∣ = v − v 0 , P K ( E 1 ) + liminf n → ∞ P K ( F ′ n ) ⩽ liminf n → ∞ P K ( F n ) .

Since E 0 is bounded, we can suppose that E 0 ∩ E 1 = ∅ . If v 1 = v − v 0 , then ∣ E 0 ∪ E 1 ∣ = ∣ E 0 ∣ + ∣ E 1 ∣ = v and by (5.4), we obtain

P K ( E 0 ∪ E 1 ) = P K ( E 0 ) + P K ( E 1 ) ⩽ P K ( E 0 ) + lim n → ∞ P K ( F n ) ⩽ I ( v ) .

Thus, E 0 ∪ E 1 is the isoperimetric region of volume v . If v 1 < v − v 0 , then E 0 ∪ E 1 is isoperimetric for its volume. Otherwise, we use Proposition 4.10 to find O ⊂ G bounded such that ∣ O ∣ = v 0 + v 1 , P K ( O ) < P K ( E 0 ) + P K ( E 1 ) and O is disjoint to any F ′ n for n big enough. Then, by (5.5),

I K ( v ) ⩽ liminf n → ∞ P K ( O ∪ F ′ n ) = P K ( O ) + liminf n → ∞ P K ( F ′ n ) < P K ( E 0 ) + P K ( E 1 ) + liminf n → ∞ P K ( F ′ n ) ⩽ P K ( E 0 ) + liminf n → ∞ P K ( F n ) ⩽ I K ( v ) ,

and we have a contradiction.

By induction, we obtain a sequence of sets E 0 , … , E n pairwise disjoint of volumes v 0 , … , v n whose union is isoperimetric for its volume ∑ i = 1 n v i . Suppose that there exists an infinite number of pieces E i . Then, ∑ i = 0 ∞ v i ⩽ v . Fixed j , we can assume without loss of generality that the family given in Theorem 1.2, ( E j ) m , and E i are pairwise disjoint for i ≠ j and small m . Let v k be small enough so that (2.18) holds and C def v k < C isop v k ( l − 1 ) ⁄ l , where C def is the constant obtained applying Theorem 1.2 to E j and C isop is given in Theorem 2.5. Hence, by (1.1) and (2.18), we obtain

I K ∑ i v i ⩽ ∑ i ≠ j , k P K ( E i ) + P K ( ( E j ) v k ) ⩽ ∑ i ≠ k P K ( E i ) + C def v k < ∑ i ≠ k P K ( E i ) + C isop v k ( l − 1 ) ⁄ l ⩽ ∑ i ≠ k P K ( E i ) + P K ( E k ) = I K ∑ i v i ,

which is a contradiction. Therefore, there are a finite number of pieces, r , until ∑ i = 1 r v i ⩾ v , and E 0 ∪ E 1 … ∪ E r is the isoperimetric region of volume v .□

Corollary 5.3

Let ( R d , · , X , K ) be a sub-Finsler nilpotent group. The isoperimetric profile I K is sub-additive.

Proof

Let v 1 , … , v n ⩾ 0 and E k an isoperimetric region of volume v k for 1 ⩽ k ⩽ n . By Theorem 4.9, E k is bounded and we can take E j ∩ E i = ∅ . Therefore,

□ I K ( v 1 + … + v n ) ⩽ P K ⋃ k = 1 n E k = ∑ k = 1 n P K ( E k ) = ∑ k = 1 n I K ( v k ) .

We prove a uniform version of the deformation lemma.

Proposition 5.4

Let 0 < a < b and denote by E v an isoperimetric region of volume a ⩽ v ⩽ b . There exist C def > 0 and λ 1 < 0 such that for any E v there is a family of sets { E v λ } λ 1 < λ with ∣ E v λ ∣ = v + λ and

(5.6) P K ( E v λ ) ⩽ P K ( E v ) + C def ∣ λ ∣ .

Proof

First, we prove that ∃ m > 0 and { p v } a ⩽ v ⩽ b such that

∣ E v ∩ B ( p v , 1 ) ∣ ⩾ m a ⩽ v ⩽ b .

Otherwise, we can take a sequence { v k } k ⊂ [ a , b ] such that

∣ E v k ∩ B ( p , 1 ) ∣ ⩽ 1 k ,

for any p ∈ R d . By Lemma 5.1 and Proposition 4.1, it holds

C 1 ⁄ l ∣ E a ∣ ⩽ C 1 ⁄ l ∣ E v k ∣ ⩽ 1 k P K ( E v k ) ⩽ 1 k P K ( E b ) ,

and ∣ E a ∣ = 0 , which is a contradiction.

As stated in Remark 3.4, given a set E v , the constant C def > 0 depends only on the radius of a ball inside E v . Therefore, it is enough to show that there exist r 0 > 0 and { p v } a ⩽ v ⩽ b such that B ( p v , r 0 ) ⊆ E v for all a ⩽ v ⩽ b . Let

r v = sup { r > 0 : B ( q v , r ) ⊆ E v for some q v ∈ E v } .

We prove that inf v ∈ [ a , b ] r v > 0 by contradiction. Otherwise, there is a sequence { v n } ⊂ [ a , b ] such that r v n < 1 n . By abuse of notation, we denote as E v n the set ℓ − p v n E v n . By Proposition 4.1, P K ( E v n ) ⩽ P K ( E b ) and E v n converges in L loc 1 ( R d ) to a finite perimeter set F . Note that ∣ E v n ∩ B ( 0 , 1 ) ∣ ⩾ m , and thus, ∣ F ∩ B ( 0 , 1 ) ∣ ⩾ m > 0 . Let p ∈ F 1 and ε > 0 as in Lemma 4.2. Fix r > 0 small enough such that

2 ε > ∣ B ( p , r ) \ F ∣ ∣ B ( p , r ) ∣ ⩾ C ∣ B ( p , r ) \ F ∣ r l .

Hence, the L loc 1 ( R d ) -convergence and Lemma 4.2 gives us that B ( p , r ⁄ 2 ) ⊆ E v n for n big enough, which is a contradiction with r v n < 1 n .□

Corollary 5.5

Let ( R d , · , X , K ) be a sub-Finsler nilpotent group. The isoperimetric profile is a locally Lipschitz function.

Proof

Fix v > 0 . Let 0 < w 0 < v , E w be the isoperimetric region of volume w 0 < w ⩽ v , and { E w λ } λ 1 < λ the family defined in Lemma 5.4. Up to taking w 0 closer to v , we can assume that λ 1 < w − v . Note that ∣ E w v − w ∣ = v and by (5.6), we obtain

I K ( v ) ⩽ P K ( E w v − w ) ⩽ P K ( E w ) + C def ( v − w ) = I K ( w ) + C def ( v − w ) .

Similarly, using that ∣ E v w − v ∣ = w , we obtain

I K ( v ) = P K ( E v ) ⩾ P K ( E v w − v ) − C def ( v − w ) ⩾ I K ( w ) − C def ( v − w ) .

By similar reasoning for 0 < v < w 0 , we obtain that I K is locally Lipschitz.□

Acknowledgments

The author would like to thank Manuel Ritoré for suggesting the problem and his help, to Gioacchino Antonelli, Simone Verzellesi, and Giacomo Vianello for fruitful conversations about the content of the manuscript and to the referee for their comments to improve the final presentation of the paper.

Funding information: The author is supported by the FEDER grant PID2023-151060NB-I00 funded by MICIU/AEI/10.13039/501100011033/ and Fundación Ramón Areces grants XXXV convocatoria para la amplación de estudios en el extranjero en ciencias de la vida y la materia.
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results, and manuscript preparation.
Conflict of interest: The author states no conflict of interest.

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Received: 2024-09-20

Revised: 2025-03-13

Accepted: 2025-04-23

Published Online: 2025-07-31

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Keywords for this article

isoperimetric regions; Nilpotent Lie groups; sub-Riemannian geometry

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