Rotational Symmetries and Spherical Measure in Homogeneous Groups

We present different types of rotational symmetries for distances in homogeneous groups, showing that the area formula for the associated spherical measure takes a simple form.


Introduction
The problem of computing the spherical measure of submanifolds in homogeneous groups still has a number of questions to be investigated. Its intriguing aspect is how it naturally involves the algebraic structure of the group and the "tangential properties" of the submanifold. The anisotropic infinitesimal behavior of the submanifold in many cases leads to new geometric questions.
Our motivations have their roots in the wider project to expand results and tools of Geometric Measure Theory in the framework of noncommutative homogeneous groups. A fundamental concept is that of area, that is obtained using the spherical measure with respect to the distance of the group. To our knowledge, the first integral formulas for this measure date back to the work of Pansu [29] in Heisenberg groups, and of Heinonen [14] in Carnot groups. Related developments appeared later in connection with the study of BV functions, sets of finite perimeter, intrinsic rectifiable sets, currents and characterizations of intrinsic regular submanifolds. Due to the large and always expanding literature, we limit ourselves to mention some related works, [1,2,4,5,[9][10][11][12]15,21,22,27,28,31] and the references therein, warning the reader that this list is very far from being complete.
In the present work, we focus our attention on the role of symmetries of distances in computing the spherical measure of C 1 smooth submanifolds. Our results constitute a continuation of those in [23], that together with this paper arise from the former preprint [24]. However our tools also apply to intrinsic regular sets, as for instance in [5] and [31]. Indeed, intrinsic regular sets might be very far from being smooth in the standard sense, [16]. For such reason, area formulas in noncommutative homogenous groups cannot rely on classical results of Geometric Measure Theory and represent a rather challenging question.
Let be an n-dimensional C 1 smooth submanifold of degree N in a homogeneous group G equipped with a homogeneous distance d. The analytic notions of degree d( ) of a submanifold and of pointwise degree d ( p) at p ∈ were introduced in [26]. We will refer to formulas (2.10) and (2.11) of [23,Sect. 2.3]. For each Borel set B ⊂ , the area formula reads as follows (1.1) The integral on the left hand side represents the intrinsic measure of , introduced in [26], see for instance [23,Definition 7.3]. The spherical measure S N is constructed by the distance d. We have denoted by β d (A p ) the spherical factor with respect to A p (Definition 1.1). The homogeneous tangent space A p at p has been introduced in [23, Definition 2.7]. From a geometric viewpoint, the spherical factor β d (S) represents the maximal area among all intersections of S with metric unit balls, that stay at a bounded distance from the origin. Our main interest is in finding those conditions on the homogeneous distance d such that β d (S) is constant as S ⊂ G varies in a specific class of subspaces. In the applications such subspaces will be the homogeneous tangent spaces of a submanifold. The next definition will state the constancy of β d with respect to a fixed family F of homogeneous subspaces. As in the previous definition, we refer to a fixed graded scalar product on G and use the notion of homogeneous subspace (Sect. 2). Definition 1.2 (Rotationally symmetric distance) Let d be a homogeneous distance on a homogeneous group G and let F be a nonempty family of homogeneous subspaces of G. We say that d is rotationally symmetric with respect to F if the spherical factor β d is a constant function on F. We denote this constant by ω d (F).
If all homogeneous tangent spaces of a submanifold are contained in a family F and d is rotationally symmetric with respect to F, then (1.1) immediately yields a neat formula for the integral representation of the spherical measure. Denoting by ω d (F) the constant spherical factor and setting S N d = ω d (F)S N , where S N is introduced in (2.6), then (1.1) yields that has the form of the standard area formula for Riemannian manifolds, [7, 3.2.46]. The simplest example of constant spherical factor is in the Euclidean space, that can be seen as a commutative homogeneous group G ≈ R q of step one, equipped with the Euclidean distance d E . The spherical factor of an n-dimensional submanifold becomes for all p ∈ and the homogeneous tangent space A p coincides with the standard tangent space. The constant ω d E (F q,n ) is the Lebesgue measure of the unit ball in R n . We have denoted by F q,n the Grassmannian of n-dimensional subspaces of G ≈ R q . As soon as we consider any commutative homogeneous group of step one, then the spherical factor need not be constant. We are actually dealing with a finite dimensional Banach space, where a suitable area formula for submanifolds was proved in [3,Theorem 4.2].
The general question about the constancy of β d is nontrivial, since it depends on the algebraic and metric structure of the group. A general statement about symmetries in homogeneous groups would involve infinitely many homogeneous groups, that are not isomorphic to each other, hence their distances are not bi-Lipschitz equivalent. The fact that we have such infinitely many "different geometries" makes general symmetry results rather challenging.
Our first case concerns rotationally symmetric distances with respect to the family N n of all n-dimensional vertical subgroups (Definition 3.1). Vertical subgroups are the homogeneous tangent spaces at some points of transversal submanifolds, [23]. As a result, such rotationally symmetric distances imply a simpler integral form for the spherical measure of transversal submaniofolds, according to Theorem 1.2 below. The family of transversal manifolds includes all hypersurfaces and the larger class of non-horizontal submanifolds, [19]. For instance, in Heisenberg groups a smooth submanifold is either Legendrian or transversal.
The rotational symmetry with respect to N n is a property of n-vertically symmetric distances (Definition 3.2) that is studied in Sect. 3. An interesting aspect of such distances is their symmetry with respect to special classes of isometries, that preserve the shape of the metric unit ball, although they are not necessarily Lie group homomorphisms. The informal justification for this resides in the blow-up process, that suggests us to look at the metric unit ball as embedded in the tangent space, where we only have a linear structure. Vertical symmetric distances somehow possess the "minimal rotational symmetries" such that the following theorem holds. Another class of rotational symmetric distances is that of multiradial distances, that are explored in Sect. 5. Such distances were used in [23] to relate the Hausdorff measure to the spherical measure of horizontal submanifolds. The next theorem provides an area formula with multiradial distance for new classes of submanifolds. The area formula (1.4) also yields a simpler form for the spherical measure of curves with respect to a multiradial distance, improving the results of [17].

Theorem 1.3 (Area formulas for multiradial distances) Let G be a homogeneous group with a multiradial distance d and let
⊂ G be a C 1 smooth n-dimensional submanifold of degree N. Suppose that one of the following conditions hold.
(1) The step of G is two, every point of maximum degree is algebraically regular and points of lower degree are S N -negligible. (2) The submanifold is one dimensional.
We denote by F the family of all homogeneous subgroups of topological dimension n and Hausdorff dimension N. Thus, in both of the previous cases, d is rotationally symmetric with respect to F and we denote by ω d (F) the constant spherical factor.
The main tool to prove this theorem is a concavity property of the area of "parallel sections" of convex sets, given in Theorem 6.3. The intersections with the metric unit ball are obtained by translated subspaces of arbitrarily fixed codimension. We believe this result is certainly well known from the Brunn-Minkowski theory of convex bodies. However, due to its importance for our purposes, we also provide its proof. Formula (1.5) not only simplifies the computation of the spherical factor, but it has also applications in the study of the equality between centered Hausdorff measure and spherical measure. This question in the setting of stratified groups was posed in [9]. The right hand side of (1.5) can be proved to represent the expression of an upper centered density, where N is the homogeneous tangent space. Then we are lead to the equality between Federer density, [21], and upper centered density [7, 2.10.19]. This in turn yields the equality between spherical measure and centered Hausdorff measure. The Q-dimensional spherical measure in G (where the Hausdorff dimension of G is Q) always coincides with the Q-dimensional centered Hausdorff measure, according to [9,Corollary 4.13]. However, in general these two measures may differ, [9,21].
As a consequence of (1.5) in the case of one codimensional vertical subgroups, combining [9, (4.22)], [9, (4.23)] and [22,Theorem 1.3], we obtain that the spherical measure equals the centered Hausodorff measure of dimension Q − 1 on all G-rectifiable sets. This result extends [9,Theorem 4.28] to all homogeneous distances whose metric unit ball is convex. Another application of Theorem 1.4 is for low codimensional H-regular surfaces in Heisenberg groups H n . When the metric unit ball is convex, formula (1.5) was used in [5] to prove that spherical measure and centered Hausdorff measure coincide on low codimensional H-regular surfaces. From [23, (1.7)], assuming that the metric unit ball is convex and combining (1.5), the centered blow-up of [25, Theorem 1.1] and [9, Theorem 3.1], we obtain the equality between centered Hausdorff measure and spherical measure also on all transversal submanifolds. Similar applications of (1.3) can be obtained for multiradial distances, starting from Theorems 5.3 and 5.5.

Basic Notions
A simply connected real and finite dimensional nilpotent Lie group can be regarded as a linear space G equipped with a Lie product and a polynomial group operation arising from the Baker-Campbell-Hausdorff formula. If we add the assumption that G = H 1 ⊕ · · · ⊕ H ι and for all i, j ≥ 0 and H l = 0 for all l > ι, then we say that G is a graded group. We have the grading The integer ι ≥ 1 is called the step of the group. We introduce dilations δ r : G → G to be linear mappings such that δ r ( p) = r i p for each p ∈ H i , r > 0 and i = 1, . . . , ι. According to this homogeneity property of dilations, elements of H i can be called homogeneous vectors, having degree i. The graded nilpotent Lie group G equipped with intrinsic dilations is called homogeneous group, [8].
We may further identify G with the tangent space T 0 G at the origin 0, getting a canonical isomorphism between H j and V j , that associates to each v ∈ H j the unique left invariant vector field X ∈ V j such that X (0) = v. If we think of G as equipped with a Lie product inducing a Lie algebra structure, we get the following Baker-Campbell-Hausdorff formula with x, y ∈ G, where c j are suitably defined polynomials, [30]. We refer to (2.2) in short as BCH formula.
In the sequel q denotes the linear dimension of G. Linear subspaces S of G that satisfy δ r (S) ⊂ S for every r > 0 are called homogeneous subspaces. It is not difficult to observe that for a homogeneous subspace S ⊂ G we have the characterizing condition where each S j is a subspace of H j . If a homogeneous subspace has in addition the structure of Lie subgroup, then it is called homogeneous subgroup.
A homogeneous distance d on a graded nilpotent Lie group G is a left invariant distance with d(δ r x, δ r y) = r d( p, q) for all p, q ∈ G and r > 0. Open and closed balls are defined respectively as follows The homogeneous norm associated to a homogeneous distance d is defined as for all x ∈ G. When the graded nilpotent Lie group is equipped with the corresponding dilations, along with a homogeneous norm, is called homogeneous group. A graded basis (e 1 , . . . , e q ) of a homogeneous group G is a basis of vectors such that is a basis of H j for each j = 1, . . . , ι, where we have set m 0 = 0. We also set m = m 1 and observe that m ι = q. A graded basis provides the associated graded coordinates x = (x 1 , . . . , x q ) ∈ R q , then defining the unique element p = q j=1 x j e j ∈ G. It is easy to realize that one can always equip a homogeneous subgroup with graded coordinates. We fix throughout the paper a graded left invariant Riemannian metric g on the homogeneous group G, that automatically induces a scalar product on T 0 G. By the identification of G with T 0 G we also have a fixed scalar product on G.
The fact that our left invariant Riemannian metric g is "graded" means that the induced scalar product on G is graded, that is, all subspaces H i with i = 1, . . . , ι are orthogonal to each other. With a slight abuse of notation, both the norm on G and the norm arising from the Riemannian metric g on tangent spaces are denoted by the same symbol | · |. We refer to the norm on G as arising from a graded scalar product, hence all subspaces H i are orthogonal to each other. Clearly when a graded scalar product is fixed, we can find a graded basis that is also orthonormal with respect to this scalar product.
We close this section introducing the Carathéodory measure, which includes both spherical measure and Hausdorff measure, that are used throughout the paper.
Let F ⊂ P(G) denote a nonempty family of closed subsets and fix α > 0. For any δ > 0 and E ⊂ G, we define When F is the family of closed balls F b , then we set to be the α-dimensional spherical measure of E. In the case F is the family of all closed sets and k ∈ {1, 2, . . . , q − 1}, for any A ⊂ G we define the Hausdorff measure where L k denotes the Lebesgue measure on R k and d E denotes the Euclidean distance on R k . The outer measure φ k δ,|·| refers to the norm | · | defined by the fixed graded scalar product on G.

Vertically Symmetric Distances
The main object of this section is the proof of Theorem 1.1. We start by recalling the notion of vertical subgroup and of transversal submanifold. More information can be found in [25] and [23,Sect. 5].
We denote by N n the family of all n-dimensional vertical subgroups.
A transversal submanifold in G is a C 1 smooth submanifold having at least one point where the homogeneous tangent space is a vertical subgroup. Denoting by n the dimension of the vertical subgroup N , we can find an explicit formula for the integer appearing in (3.1). We denote this integer by n since it depends on the dimension of N . One can easily check that An equivalent definition for n is to see it as the minimal degree of homogeneous vectors belonging to N . Associated to n , we also introduce the integer where 1 ≤ n ≤ q and clearly by definition r n ≥ 1. We finally observe that the integer precisely corresponds to the Hausdorff dimension of N with respect to d. It can be proved that (3.4) is the Hausdorff dimension of any transversal submanifold of dimension n, see [25].
To introduce the notion of n-vertically symmetric distance, we define the following class of isometries where we have taken into account that the subspaces H i and H j of G are orthogonal for i = j and F is a family of isometries of H n . We also introduce the subspaces where U n = {0} for n = 1.
Definition 3.2 (n-vertically symmetric distance) Let 1 ≤ n ≤ q and consider n as in (3.2) and r n as in (3.3). We introduce the integer If j = 0, then we say that d is n-vertically symmetric. In the case j > 0, we say that d is n-vertically symmetric if the following conditions hold.
• There exists a family F of isometries of H n such that for every couple of jdimensional subspaces S 1 , S 2 ⊂ H n , there exists J ∈ F that satisfies • Let U n and V n be as in (3.6) and let P V n : G → V n and P U n : G → U n be the orthogonal projections onto V n and U n , respectively. We consider O n (F) defined in (3.5) by our family of isometries F. Then we have: The metric unit ball B(0, 1) is with respect to d and we have assumed that when n = 1, then ψ is any monotone nondecreasing function from [0, +∞) to [0, +∞).

Remark 3.1
The previous definition when n = q − 1, n = 1 and ψ(t) = t, yields the notion of H 1 -vertical symmetry introduced in [22, Definition 6.1]. In this definition the transitivity of isometries of H 1 corresponds to (3.7) for j = dim H 1 − 1. In a few words, any H 1 -vertically symmetric distance is also a (q − 1)-vertically symmetric distance. [6]. It can be checked that d is p-vertically symmetric for any p = 1, . . . , dim G − 1. This will be also a consequence of Proposition 5.1.

Example 3.2 In any H-type group with direct decomposition
The next lemma provides a factorization of a homogeneous group G with respect to a vertical subgroup N , although a complementary subgroup V such that V ⊕ N = G may not exist. We simply consider V as a homogeneous subspace of G.

Lemma 3.3
If G is a homogeneous group, V ⊂ G is a homogeneous subspace and N ⊂ G is a vertical subgroup such that V ⊕ N = G, then the mapping is an analytic diffeomorphism. Furthermore, its inverse mapping T : Proof The mapping F(v, h) = vh has the property that so the assumption that V ⊕ N = G implies that d F(0, 0) is invertible and F is locally invertible around the origin. The homogeneity of F, i.e.
shows that F is surjective. We now consider where v, w ∈ V and h, k ∈ N . By the BCH formula (2.2), we have Applying P V to the equality (3.8), it follows that We have shown that any element x ∈ G can be uniquely written as the product therefore concluding the proof.

Lemma 3.4 If G is a homogeneous group, V ⊂ G is a homogeneous subspace and
Proof From (2.2) and the fact that N is an ideal with respect to the Lie algebra structure of G, there holds considering v + n ∈ G and applying Lemma 3.3 we get We have then established the opposite inclusion.
An important feature of vertical subgroups is the following left invariance property of the Hausdorff measure, once we consider our fixed graded scalar product on G. x j e j and n = n j=1 ζ j e q−n+ j ∈ N .
We express the left translation explicitly (3.11) According to the Baker-Campbell-Hausdorff formula, the functions c j are polynomials that only depends on variables of degree less than d j . We have defined where (b 1 , . . . ,b q ) is the canonical basis of R q . Let us remark thatζ d j −1 = 0 for d j ≤ d q−n+1 . We decompose p into the sum x j e j and w = q j=q−n+1 x j e j .
With this notation formula (3.9) gives pN = v + N . Since v / ∈ N , this decomposition improves (3.11), giving with k n = max {q − n + 1, m + 1}. We now consider the projection . . . , b n ) is the canonical basis of R n and J : R n → N , defined as The composition F : R n → R n defined as F = T • l p • J can be written as follows As a consequence of the previous formulas, for every j, l = 1, . . . , n we get In the special case l ≥ j, due to (3.12) the function ζ → c q−n+ j (x q−n+ j , ζ ) only depends on ζ i with d q−n+i ≤ d q−n+ j − 1, therefore i < j ≤ l. We have proved that hence the Jacobian of F is one. Since both T and J are isometries one easily observes that image measures satisfy T H n |·| = L n and J L n = H n |·| .
Since F preserves the Lebesgue measure L n , the following equalities conclude the proof, that is The previous results allow us to establish our first main result.

Proof of Theorem 1.1
First of all, we consider the integers n and r n defined in (3.2) and (3.3), respectively. If r n = dim H n , then j = 0 in Definition 3.2 and d is automaticaly n-vertically symmetric. Thus, the only n-dimensional vertical subgroup is N 0 = H n ⊕ · · · ⊕ H ι and the spherical factor is obviously constant and equal to β(d, N 0 ). Let us consider the case 0 < j = r n < dim H n , where n = j + h n +1 + · · · + h ι .
To simplify notation, in the rest of the proof we will write in place of n . We fix z ∈ B(0, 1) and consider two arbitrary n-dimensional vertical subgroups where S 1 and S 2 are j -dimensional subspaces of H and By the n-vertical symmetry of d, there exists an isometry J : This defines the isometry T : G → G such that for each j = . We now define the subspace such that Z 1 is orthogonal to S 1 and We consider the orthogonal projection P V 1 : G → V 1 , that is also the linear projection associated to the direct sum Defining the nonlinear projection
Since the previously defined mapping T belongs to O, in view of the property (2) of Definition 3.2, we obtain that It is not difficult to realize that due to the definition of T and the inclusions S 1 ⊂ H . We have proved that (3.14) We wish to check whether (T v 1 ) −1 is a suitable element of B(0, 1). To do this, we define the subspace V = H 1 ⊕ · · · ⊕ H and consider the orthogonal decompositions 1 and v are orthogonal to η 1 ∈ N 1 and η ∈ W , respectively. The previous equality gives hence v 1 and s 1 are orthogonal. We write the orthogonal decomposition v 1 = w 1 + z 1 with w 1 ∈ H 1 ⊕ · · · ⊕ H −1 and z 1 ∈ Z 1 , therefore v = w 1 + z 1 + s 1 . Since s 1 is orthogonal to w 1 and to z 1 , we get |P H v| = |z 1 + s 1 | = |z 1 | 2 + |s 1 | 2 ≥ |z 1 |.

By the property (1) of Definition 3.2, since
we have ψ(u, t) ≥ 0 that is monotone nondecreasing with respect to t and such that We have proved that v 1 ∈ V ∩ B(0, 1). Moreover, taking into account that It follows that and clearly v 0 = (T v 1 ) −1 ∈ V ∩ B(0, 1). Due to (3.10), it follows that Since T is an isometry, the images T v 1 and N 2 = T (N 1 ) are orthogonal, being so v 1 and N 1 . We may apply (3.9), getting The previous equality joined with (3.14) yields and the arbitrary choice of z ∈ B(0, 1) yields β d (N 1 ) ≤ β d (N 2 ). Exchanging the role of N 1 with that of N 2 , we are able to conclude the proof.

Remark 3.6
The intriguing aspect of Theorem 1.1 is that the metric unit ball is not assumed to satisfy a strong geometric condition like convexity. The role of convexity is specifically studied in Sect. 6.
As an example, the sub-Riemannian ball in the Heisenberg group is not convex, but it is 2-vertically symmetric, as already pointed out in [22]. This is also an interesting case where a distance is vertical symmetric, but it does not satisfy the stronger condition of being multiradial. Multiradial distances will be studied in Sect. 5.

Area and Coarea Formulas in Low Codimension
The area formula In this section we always assume k ≤ m. Then a C 1 smooth submanifold of codimension k is non-horizontal if and only if its degree is Q − k. We fix a Riemannian metricg such that its associated volume measure volg is left invariant. Considering also our fixed left invariant Riemannian metric g, we introduce the associated the linear isomorphismsg * , g * : T G → T * G. In a canonical way, taking wedge products, such mappings generateg * k , g * k : k (T G) → k (T G). Following the notation of [19], we introduce the unit k-normals n andñ of a non-horizontal submanifold , that defined with respect to the Riemannian metrics g andg, respectively. The horizontal k-normal at p with respect tog and g is defined as for every p ∈ of degree Q − k. Using the horizontal projection π g,H with respect to g ( [19, Definition 3.1]), we set n g,H ( p) = π g,H (ñ g ( p)).
We define the unique geometric constant c(g,g) > 0 such that Since the class of (q − k)-vertically symmetric distances is larger than the one in [19, Sect. 6], using Theorem 1.2 we obtain an area formula with constant spherical factor for a larger family of distances.  (N q−k ) be the constant spherical factor associated to this family. We fix any Riemannian metricg whose volume measure is left invariant. We rescale the spherical measure as follows

For every Borel set B ⊂ , it follows that
where Q n is given by formula (3.4).
Our claim is then a consequence of (1.3) joined with formula (12) of [19].
The spherical measure S Q−k appearing in (4.1) is introduced at the end of Sect. 2.

Remark 4.2
It is well known that any C 1 smooth hypersurface ⊂ G is automatically a non-horizontal submanifold, therefore (4.2) holds for any C 1 smooth hypersurface of a homogeneous group G. The local isoperimetric inequality applied to a suitably "small" open subset U ⊂ of shows that it must have positive S Q−1 measure. Since characteristic points are S Q−1 negligible, [18], the subset U must contain noncharacteristic points. This shows that has degree Q − 1.
In any homogeneous group G equipped with graded coordinates x j , the corresponding basis of left invariant vector fields X 1 , . . . , X q has the form and it is automatically assumed to be orthonormal with respect to the fixed left invariant metric g. The dual basis of left invariant differential forms has the form The special form of the left invariant differential forms ξ j implies that ξ 1 ∧ ξ 2 ∧ · · · ∧ ξ q = dx 1 ∧ dx 2 ∧ · · · ∧ dx q , therefore using the Euclidean metric in place ofg yields This simplifies the expression of (4.2), when use the Euclidean metric to compute the spherical measure of a submanifold.

Example 4.3
Using graded coordinates x j in a homogeneous group G, we consider the standard Euclidean metricg given by δ i j , along with our fixed left invariant Riemannian metric g. We notice that the volume measure volg associated tog is left invariant. Let be a non-horizontal submanifold of codimension k, with unit normalñ E with repect to the Euclidean metricg. We define k-vector hence [19,Proposition 3.1] gives where ·, · E denotes the Euclidean scalar product on k-vectors and n E,g,H = π g,H ñ E,g .
Since in this case c(g,g) = 1, combining (4.1) and (4.2), we obtain for every Borel set B ⊂ and for any (q − k)-vertically symmetric homogeneous distances d. The symbol H q−k E denotes the (q − k)-dimensional Hausdorff measure with respect to the Euclidean distance. In particular, since all smooth hypersurfaces are non-horizontal submanifolds, the previous formula for k = 1 yields the well known formula for the spherical measure S Q−1 of hypersurfaces that is here extended to the largest class of suitably symmetric distances, namely the (q − 1)-vertically symmetric distances. In this caseñ E is a normal to with respect to the Euclidean metric.
for any nonnegative measurable function u : where vol g is the Riemannian volume measure on G and is the horizontal Jacobian at every differentiability point x ∈ A of f .
We point out that the class of rotationally symmetric distances for which the coarea formula (4.5) holds is larger than the family of distances considered in [19]. The arguments to establish the previous corollary are the same ones of [19,Theorem 1.2]. In our case they are combined with the area formula (4.2). If we divide both members of (4.5) by c(g,g), we immediately establish that

Multiradial Distances
In this section, we study integral formulas for the spherical measure with respect to a multiradial distance, that is another type of "rotational symmetric distance". Two important classes of multiradial distances are the Cygan-Korányi distance (Example 3.2) and the d ∞ distance (Example 6.2).

Definition 5.1 (Multiradial distance)
We say that a homogeneous distance d on a homogeneous group G is multiradial if there exists ϕ : [0, +∞) ι → [0, +∞) continuous and monotone nondecreasing on each single variable, such that x j = P H j (x) and P H j : G → H j is the canonical projection with respect to the direct sum decomposition of G into subspaces H j .
As an easy consequence of 1-homogeneity of d with respect to dilations, one may also observe that ϕ in the definition of multiradial distance satisfies ϕ(x) → +∞ as |x| → +∞, where | · | is the Euclidean norm of R ι .
The following proposition shows that the assumptions of Theorem 1.1 also hold when the distance d is multiradial.
Proof We represent the metric unit ball as follows observing that in general it need not be convex. Let us fix = n , referring to (3.2). Let us consider the nontrivial case where n = j + h +1 + · · · + h ι and 0 < j < h , we consider two j-dimensional subspaces S 1 , S 2 ⊂ H . We choose any isometry J : H → H with J (S 1 ) = S 2 with respect to the fixed graded scalar product on G and set F to be the set of all isometries of H . The map T : G → G defined as for all j = , is clearly an isometry of G. The special representation (5.2) of the metric unit ball clearly gives T (B(0, 1)) = B(0, 1).
Let us consider the projection P V : G → V with V = H 1 ⊕ · · · ⊕ H and observe that in view of the nondecreasing monotonicity of ϕ with respect to each single variable. Clearly, any element v ∈ B(0, 1) ∩ V is equal to P V (v), therefore (B(0, 1)).
If we set ψ : then also property (1) of Definition 3.2 is established, that is d is an n-vertically symmetric distance.

Remark 5.2
As a consequence of Proposition 5.1 joined with Theorem 1.1, the area formulas (1.3) and (4.2), and the coarea formula (4.5) hold with respect to any multiradial distance.

Multiradial Distances in Step Two Homogeneous Groups
The next theorem shows that in step two homogeneous groups the spherical factor with respect to a multiradial distance has a simple formula. The same formula will be obtained in Sect. 6 for all homogeneous distances such that their metric unit ball is convex.

Theorem 5.3
If G is a step two homogeneous group and d is a multiradial distance, then for every n-dimensional homogeneous subspace V ⊂ G we have where 1 ≤ n ≤ q − 1 and B = {x ∈ G : d(x, 0) ≤ 1}.
Proof We choose z ∈ B and write V = V 1 ⊕ V 2 with V j ⊂ H j , being V a homogeneous subspace of G. Then the assumptions on d ensure that B is defined as in (5.2), therefore we get The BCH formula (2.2) yields . From the coercivity of ϕ we can define then considering an orthogonal system of coordinates on V and denoting by L n the corresponding Lebesgue measure on V , Fubini's theorem yields where B E (z, r ) = {x ∈ G : |x − z| < r } and n j = dim V j with j = 1, 2. From the nondecreasing monotonicity of ϕ with respect to each variable, defining whenever |x| < r 1 gives the monotonicity As a consequence, using the integral representation (5.4) for z = 0 we get the formula Taking into account the last inequality, Theorem 6.3 and formula (5.6), we get therefore concluding the proof.
By (5.3) we show that a multiradial distance is also rotationally symmetric with respect to "large families" of homogeneous subspaces.

Theorem 5.4
Let G be a homogeneous group of step two and let d be a multiradial distance. Let 1 ≤ n 1 , n 2 ≤ q − 1 be two integers such that n 1 + n 2 = n ≤ q − 1 and let F n 1 ,n 2 be the family of all homogeneous subspaces V = V 1 ⊕ V 2 ⊂ G such that dim V 1 = n 1 and dim V 2 = n 2 . Then d is rotationally symmetric with respect to F n 1 ,n 2 .
Proof Since G is of step two, we have G = H 1 ⊕H 2 . Let us consider two homogeneous subspaces V and W with direct decompositions V 1 ⊕ V 2 and W 1 ⊕ W 2 , respectively, where V 1 , W 1 ⊂ H 1 , V 2 , W 2 ⊂ H 2 and we have the conditions dim V 1 = dim W 1 = n 1 and dim V 2 = dim W 2 = n 2 .
Considering two isometries J 1 : H 1 → H 1 and J 2 : H 2 → H 2 such that respectively, then for every x 1 ∈ H 1 and x 2 ∈ H 2 we define The mapping T : G → G is an isometry, since H 1 is orthogonal to H 2 with respect to our fixed graded scalar product on G. Moreover, the shape of the metric unit ball B gives the equalities As a consequence, formula (5.3) concludes the proof.

One Dimensional Rotational Symmetries
We show that any multiradial distance has rotational symmetries with respect to one dimensional homogeneous subspaces of the Lie algebra.
Theorem 5.5 Let d be a multiradial distance on a homogeneous group G and let 1 ≤ N ≤ ι. Then for every one dimensional subspace L ⊂ H N we have Proof We consider a one dimensional subspace L of H N . We observe that Since d is multiradial, we may write The nondecreasing monotonicity of ϕ with respect to each variable shows that Observing that we have shown our claim.
so observing that the subspaces H i are orthogonal we conclude that T : G → G is an isometry. The fact that d is multiradial implies that As a consequence, formula (5.7) concludes the proof.

Remark 5.7
Let us point out that all one dimensional subspaces of a homogeneous group are automatically Lie subgroups.

Proof of Theorem 1.3 and Applications
In this section we apply the previous results to obtain a standard form of area formulas with respect to multiradial distances. Throughout this section, G is a fixed homogeneous group and ⊂ G is an n-dimensional C 1 smooth submanifold of degree N.
Proof of Theorem 1. 3 Let us now assume that conditions (1) hold. If V ∈ F is any homogeneous subgroup of the form V = V 1 ⊕ V 2 , with dim V 1 = n 1 and dim V 2 = n 2 , then we must have n 1 + n 2 = n and n 1 + 2n 2 = N.
The previous conditions uniquely define the integers n 1 = 2n − N and n 2 = N − n. A simple application of the previous corollary is given in the next example.

Example 5.9
Let us consider an open, bounded and connected set U ⊂ R n , assuming that there exists another open setŨ with U ⊂Ũ and a smooth embedding :Ũ → G of class C 1,1 . We assume that G is equipped with a structure of H-type group equipped with a multiradial distance that satisfies for all x ∈ G, see Example 3.2. Let | · | be the norm on G associated to this H-type structure. We consider the C 1,1 submanifold = (U ) and denote its degree by N. Therefore we have two uniquely defined integers n 1 = 2n − N and n 2 = N − n that satisfy n = n 1 + n 2 and N = n 1 + 2n 2 . By Theorem 5.4, d is rotationally invariant with respect to the family F 2,G of all n-dimensional homogeneous subgroups V of Hausdorff dimension N. We denote by ω d (F 2,G ) the associated constant spherical factor and introduce the renormalized spherical measure S N d = ω d (F 2,G )S N . We denote by y = (y 1 , . . . , y n ) the local coordinates of with respect to . As a consequence of Corollary 5.8 and [23, Proposition 7.5], we get where g is the left invariant Riemannian metric associated to fixed norm | · | on G.

Remark 5.10
Assuming that we have a transversal C 1 smooth curve in the Heisenberg group H n , namely a non-horizontal curve, then Theorem 1.2 yields the area formula for any homogeneous distance. We do not need further assumptions on the distance. From Definition 3.2 one may easily notice that in H n any homogeneous distance is automatically 1-vertically symmetric. In particular, formula (5.9) holds for the sub-Riemannian distance of H n , that is not multiradial.

Sections of Convex Balls
In this section, we study the spherical factor of those homogeneous distances whose metric unit ball is a convex set. Such distances can be found in any homogeneous group, according to the next examples.
Example 6.1 Theorem 2 of [13] shows that in a homogeneous group we can find a homogeneous distance whose metric unit ball is an Euclidean ball with suitably small radius, where all layers H j in the decomposition of G are orthogonal. In particular, the metric unit ball is a convex set.

Example 6.2
The homogeneous distance d ∞ of [10] can be easily checked to have a convex metric unit ball. This distance is also multiradial.
The Cygan-Korányi distance of Example 3.2 has a convex metric unit ball and it is multiradial. The following fact is our crucial tool. Our point is to use the the classical Brunn-Minkowski inequality, in dimension n. We observe the following equality of sets for every γ ∈ R and u ∈ H , hence we define Thus, the inclusion (6.1) can be written as follows It follows that We are now in the position to prove an explicit formula for the spherical factor when the metric unit ball is convex.
Taking into account (6.5) and (6.8), we have proved that Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.