Rigidity for perimeter inequality under spherical symmetrisation

Necessary and sufficient conditions for rigidity of the perimeter inequality under spherical symmetrisation are given. That is, a characterisation for the uniqueness (up to orthogonal transformations) of the extremals is provided. This is obtained through a careful analysis of the equality cases, and studying fine properties of the circular symmetrisation, which was firstly introduced by Pólya in 1950.


Introduction
In this paper we study the perimeter inequality under spherical symmetrisation, giving necessary and sufficient conditions for the uniqueness, up to orthogonal transformations, of the extremals. Perimeter inequalities under symmetrisation have been studied by many authors, see for instance [19,20] and the references therein. In general, we say that rigidity holds true for one of these inequalities if the set of extremals is trivial. The study of rigidity can have important applications to show that minimisers of variational problems (or solutions of PDEs) are symmetric.
For instance, a crucial step in the proof of the Isoperimetric Inequality given by Ennio De Giorgi consists in showing rigidity of Steiner's inequality (see, for instance, [21,Theorem 14.4]) for convex sets (see the proof of Theorem I in Section 4 in [15,16]). After De Giorgi, an important contribution in the understanding of rigidity for Steiner's inequality was given by Chlebík, Cianchi, and Fusco. In the seminal paper [11], the authors give sufficient conditions for rigidity which are much more general than convexity. After that, this result was extended to the case of higher codimensions in [2], where a quantitative version of Steiner's inequality was also given.
Then, necessary and sufficient conditions for rigidity (in codimension 1) were given in [8], in the case where the distribution function is a Special Function of Bounded Variation with locally finite jump set [8,Theorem 1.29]. The anisotropic case has recently been considered in [25], where rigidity for Steiner's inequality in the isotropic and anisotropic setting are shown to be equivalent, under suitable conditions. In the Gaussian setting, where the role of Steiner's inequality is played by Ehrhard's inequality (see [14,Section 4.1]), necessary and sufficient conditions for rigidity are given in [9], by making use of the notion of essential connectedness [9,Theorem 1.3]. Finally, in the smooth case, sufficient conditions for rigidity are given in [23,Proposition 5], for a general class of symmetrisations in warped products. For the study of rigidity of functional inequalities we refer the reader to [8,11,13,15].
The main motivation for the study of the spherical symmetrisation is that it can be used to understand the symmetry properties of the solutions of PDEs and variational problems, when the radial symmetry has been ruled out. Moreover, some well established methods (as for instance the moving plane method, see [18,28]) rely on convexity properties of the domain which fail, for instance, when one deals with annuli.
In particular, in many applications minimisers of variational problems and solutions of PDEs turn out to be foliated Schwarz symmetric. Roughly speaking, a function u : R n → R is foliated Schwarz symmetric if one can find a direction p ∈ S n−1 such that u only depends on |x| and on the polar angle α = arccos(x · p), and u is non increasing with respect to α (herex := x/|x|, and | · | denotes the Euclidean norm in R n ). We direct the interested reader to [3][4][5]30] and the references therein for more information.

Spherical symmetrisation
To the best of our knowledge, the spherical symmetrisation was first introduced by Pólya [26], in the case n = 2 and in the smooth setting. Let n ∈ N with n ≥ 2. For each r > 0 and x ∈ R n , we denote by B(x, r ) the open ball of R n of radius r centred at x, by ω n the (n-dimensional) volume of the unit ball, and we write B(r ) for B(0, r ). Moreover, e 1 , . . . , e n stand for the vectors of the canonical basis of R n . Given a set E ⊂ R n and r > 0, we define the spherical slice E r of E with respect to ∂ B(r ) as Let v : (0, ∞) → [0, ∞) be a measurable function. We say that for H 1 -a.e. r ∈ (0, ∞), (1.1) Let r > 0, p ∈ S n−1 , and β ∈ [0, π] be fixed. The open geodesic ball (or spherical cap) of centre r p and radius β is the set B β (r p) := {x ∈ ∂ B(r ) : dist S n−1 (x, p) < β}.
The (n − 1)-dimensional Hausdorff measure of B β (r p) can be explicitly calculated, and is given by Among all the spherically v-distributed sets of R n , we denote by F v the one whose spherical slices are open geodesic balls centred at the positive e 1 axis., i.e.

Basic notions on sets of finite perimeter
Let E ⊂ R n be a measurable set, and let t ∈ [0, 1]. We denote by E (t) the set of points of density t of E, given by The essential boundary of E is then defined as Moreover, if A ⊂ R n is any Borel set, we define the perimeter of E relative to A as the extended real number given by and we set P(E) := P(E; R n ). When E is a set with smooth boundary, it turns out that ∂ e E = ∂ E, and the perimeter of E agrees with the usual notion of (n − 1)-dimensional surface measure of ∂ E.
If P(E) < ∞, it is possible to define the reduced boundary ∂ * E of E. This has the property that ∂ * E ⊂ ∂ e E, H n−1 (∂ e E \ ∂ * E) = 0, and is such that for every x ∈ ∂ * E there exists the measure theoretic outer unit normal ν E (x) of ∂ * E at x, see Sect. 2. If x ∈ ∂ * E, it will be convenient to decompose ν E (x) as  where ν E ⊥ (x) := (ν E (x) ·x)x and ν E (x) are the radial and tangential component of ν E (x) along ∂ B(|x|), respectively. In the following, we will use the diffeomorphism : (0, ∞) × S n−1 → R n \ {0} defined as (r , ω) := r ω for every (r , ω) ∈ (0, ∞) × S n−1 .

Perimeter inequality under spherical symmetrisation
Our first result shows that the spherical symmetrisation does not increase the perimeter, and gives some necessary conditions for equality cases. In our analysis we require the set F v (or, equivalently, any spherically v-distributed set) to have finite volume. This is not restrictive. Indeed, if F v has finite perimeter but infinite volume, we can consider the complement R n \ F v which, by the relative isoperimetric inequality, has finite volume. This change corresponds to considering the complementary distribution function r → nω n r n−1 − v(r ), and the spherical symmetrisation with respect to the axis −e 1 .
The result above shows that the perimeter inequality holds on a local level, provided one considers sets of the type (B × S n−1 ), with B ⊂ (0, ∞) Borel. Inequality (1.4) is very well known in the literature. In the special case n = 2, a short proof was given by Pólya in [26]. In the general n-dimensional case with B = (0, ∞) the result is stated in [24, Theorem 6.2]), but the proof is only sketched (see also [22] and [23, Proposition 3 and Remark 4]). As mentioned by Morgan and Pratelli in [24], certain parts of the proof of ( We start by introducing radial and tangential components of a Radon measure, see Sect. 3.1. These turn out to be useful tools which allow to prove several preliminary results. Moreover, since we are dealing with a symmetrisation of codimension n − 1, we need to pay attention to some delicate effects that are not usually observed when the codimension is 1 (as, for instance, in [11]). Indeed, a crucial role is played by the measure λ E given by: for every Borel set B ⊂ (0, ∞). When n = 2, it turns out that λ E is singular with respect to the Lebesgue measure in (0, ∞). However, for n > 2 it may happen that λ E contains a non trivial absolutely continuous part, see Remark 3.9. This requires some extra care while proving inequality (1.4). A similar phenomenon has already been observed in [2], in the study of the Steiner symmetrisation of codimension higher than 1. Higher codimension effects play an important role also in the study of rigidity, as explained below.

Rigidity of the perimeter inequality
Given v : (0, ∞) → [0, ∞) measurable, satisfying (1.2), and such that F v is a set of finite perimeter and finite volume, we define N (v) as the class of extremals of (1.4): Note that, by definition of F v , and by the invariance of the perimeter under rigid transformations, every time we apply an orthogonal transformation to F v we obtain a set that belongs to N (v), i.e.: The set E above cannot be obtained by applying an orthogonal transformation to the set F v shown in the right, therefore rigidity (R) fails. This happens because the set {0 < α v < π} is disconnected by a point r ∈ (0, ∞) such that α v (r ) = π that is, when the class of extremals of (1.4) is just given by rotated copies of F v . We will say that rigidity holds true for inequality (1.4) if In order to explain which conditions we should expect in order (R) to be true, let us first give some examples. Figure 2 shows a set E ∈ N (v) that cannot be obtained by applying a single orthogonal transformation to F v . This is due to the fact that the set {0 < α v < π} is disconnected by a pointr satisfying α v (r ) = 0. A similar situation happens when {0 < α v < π} is disconnected by points belonging to the set {α v = π}, see Fig. 3.
One possibility to avoid such a situation could be to request the set {0 < α v < π} to be an interval. However, this condition depends on the representative chosen for α v , while the perimeters of the sets E and F v don't. Indeed, in Fig. 2 one could modify α v just at the point r , in such a way that {0 < α v < π} becomes an interval. Nevertheless, rigidity still fails, see Fig. 4.
To formulate a condition which is independent on the chosen representative, we consider the approximate liminf and the approximate limsup of α v , which we denote by α ∧ v and α ∨ v , respectively (see Sect. 2). These two functions are defined at every point r ∈ (0, ∞) and In addition, they do not depend on the representative chosen for α v , and r x 1 Modifying the function α v given in Fig. 2 at the pointr , we can make sure that {0 < α v < π} is an open connected interval. However, rigidity still fails 5 An example in which rigidity fails. In this case, the tangential part of ∂ * F v gives a non trivial contribution to P(F v ). This allows to slide a proper subset of F v around the origin, without modifying the perimeter . The condition that we will impose is then the following: One can check that in the example given in Fig. 4 this condition fails, since Let us show that, even imposing (1.6), rigidity can still be violated. In the example given in Fig. 5, there is some radius r ∈ {0 < α ∧ v ≤ α ∨ v < π} such that the boundary of F v contains a non trivial subset of ∂ B(r ). In this way, it is possible to rotate a proper subset of F v around the origin, without affecting the perimeter. Note that at each point of the set ∂ * F v ∩ ∂ B(r ) the exterior normal ν F v is parallel to the radial direction. To rule out the situation described in Fig. 5, we will impose the following condition: Note that, from Theorem 1.1 and identity (1.3), it follows that in general we only have α v ∈ BV loc (0, ∞). However, it turns out that (1.7) is equivalent to ask that 3. Our main result shows that the two conditions above give a complete characterisation of rigidity for inequality (1.4) (below,I stands for the interior of the set I).
Theorem 1.2 Let v : (0, ∞) → [0, ∞) be a measurable function satisfying (1.2) such that F v is a set of finite perimeter and finite volume, and let α v be defined by (1.3). Then, the following two statements are equivalent: Let us point out that, although similar results in the context of Steiner and Ehrhard's inequalities already appeared in [8,9], the proof of Theorem 1.2 cannot simply use previous ideas, especially in the implication (i) ⇒ (ii). We cannot rely, as in [8], on a general formula for the perimeter of sets E satisfying equality in (1.4). Instead, we exhibit explicit counterexamples to rigidity, whenever one of the assumptions in (ii) fails. This requires a careful analysis of the transformations that one can apply to the set F v , without modifying its perimeter. This turns out to be non trivial, especially if one assumes Dα v to have a non zero Cantor part (see Proposition 8.4). Also the proof of the implication (ii) ⇒ (i) presents some difficulties. In the context of Steiner symmetrisation, this has been proved in [11,Theorem 1.3] and [2, Theorem 1.2], for codimension 1 and for every codimension, respectively. In the smooth case, a proof is given in [23,Proposition 5], for the general class of symmetrisations in warped products. For the spherical setting without any smoothness assumption, this implication has already been stated in [24,Theorem 6.2], but the proof is only sketched. A rigorous proof of this fact turns out to be more delicate than one would expect, and relies on the following result. Then, Viceversa, let (1.9) be satisfied, and suppose that P(E; (I ×S n−1 )) = P(F v ; (I ×S n−1 )). Then, (1.8) holds true.
A direct proof of Lemma 1.3 does not seem to be obvious, due to the fact that, as pointed out above, the measure λ E defined in (1.5) can have an absolutely continuous part when n > 2.
In the context of Steiner symmetrisation of higher codimension, a result playing the role of Lemma 1.3 (see [2,Proposition 3.6]) is proved using the fact that the statement holds true in codimension 1, see [11,Proposition 4.2]. For this reason, we are led to consider the circular symmetrisation, which is the codimension 1 version of the spherical symmetrisation, and was originally introduced by Pólya in the case n = 3 (see [26]). Note that, when n = 2, spherical and circular symmetrisation coincide.

Circular symmetrisation
In order to introduce the circular symmetrisation, let us first observe how the spherical symmetrisation operates on a given set E, in the special case n = 2. In this situation, for each r > 0 one intersects E with the circle ∂ B(r ) of radius r centred at the origin. Then, the symmetric set F v is obtained by centring, for each r > 0, an open circumference arc of length H 1 (E ∩ ∂ B(r )) at the point re 1 . When n > 2 one can proceed in a similar way, by first slicing the set E with parallel planes, and then by symmetrising it (in each plane) with the procedure just described. Note that, in this case, one needs to specify both the direction along which the open arcs are centred, and the direction along which the slicing through planes is performed. Let us then choose an ordered pair of orthogonal directions in R n , which we will assume to be (e 1 , e 2 ) (we will be centring open circumference arcs along e 1 , while we will be slicing the set E with parallel planes that are orthogonal to e 2 ). In the following, for each x = (x 1 , . . . , x n ) ∈ R n , we will write x = (x 12 , x ), where x 12 = (x 1 , x 2 ) ∈ R 2 and x = (x 3 , . . . , x n ) ∈ R n−2 . When x 12 = 0, we setx 12 := x 12 /|x 12 |. For each given z ∈ R n−2 , we denote by z the two-dimensional plane defined by Given a set E ⊂ R n and (r , z ) ∈ (0, ∞) × R n−2 , we define the circular slice E (r ,z ) If is a circular distribution, then for Among all the sets in R n that are circularly -distributed, we denote by F the one whose circular slices are open circumference arcs centred at the positive e 1 axis. That is, we set In the following, we introduce the diffeomorphism 12 : (0, ∞)×R n−2 ×S 1 → R n \{x 12 = 0} given by 12 We can now state a result that plays the role of Theorem 1.1 for the circular symmetrisation. (a) E (r ,x ) is H 1 -equivalent to a circular arc and ∂ * (E (r ,x ) ) = (∂ * E) (r ,x ) ; In the smooth setting and in the case n = 3, inequality (1.11) was proved by Pólya. The following result is the counterpart of Lemma 1.3 in the context of circular symmetrisation. Then, Viceversa, let (1.13) be satisfied, and suppose that P(E; (I × S 1 )) = P(F ; (I × S 1 )). Then, (1.12) holds true.
Once Lemma 1.5 is established, we can show Lemma 1.3 through a slicing argument. Finally, the proof of (ii) ⇒ (i) is concluded by showing that, if E satisfies equality in (1.4), the function associating to every r ∈ (0, ∞) the center of E r (see (7.1)) is W 1,1 loc and, ultimately, constant (see Sect. 7).
The paper is divided as follows. Section 2 contains basic results of Geometric Measure Theory that are extensively used in the following. In Sect. 3 we give the setting of the problem and introduce useful tools to deal with the spherical framework. Section 4 is devoted to the study of the properties of the functions v and ξ v , while Theorem 1.1 is proven in Sect. 5. Important properties of the circular symmetrisation are discussed in Sect. 6, where we also give the proof of Lemma 1.3. The implications (ii) ⇒ (i) and (i) ⇒ (ii) of Theorem 1.2 are proven in Sects. 7 and 8, respectively.

Basic notions of geometric measure theory
In this section we introduce some tools from Geometric Measure Theory. The interested reader can find more details in the monographs [1,17,21,29]. For n ∈ N, we denote with S n−1 the unit sphere of R n , i.e.
where | · | stands for the Euclidean norm, and we set R n 0 := R n \ {0}. For every x ∈ R n 0 , we writex := x/|x| for the radial versor of x. We denote by e 1 , . . . , e n the canonical basis in R n , and for every x, y ∈ R n , x · y stands for the standard scalar product in R n between x and y. For every r > 0 and x ∈ R n , we denote by B(x, r ) the open ball of R n with radius r centred at x. In the special case x = 0, we set B(r ) := B(0, r ). In the following, we will often make use of the diffeomorphism : (0, ∞) × S n−1 → R n 0 defined as (r , ω) := r ω for every (r , ω) ∈ (0, ∞) × S n−1 .
For x ∈ R n and ν ∈ S n−1 , we will denote by H + x,ν and H − x,ν the closed half-spaces whose boundaries are orthogonal to ν: is a sequence of Lebesgue measurable sets in R n with finite volume, and E ⊂ R n is also measurable with finite volume, we say that {E h } h∈N converges to E as h → ∞, and write In the following, we will denote by χ E the characteristic function of a measurable set E ⊂ R n .

Density points
Let E ⊂ R n be a Lebesgue measurable set and let x ∈ R n . The upper and lower n-dimensional densities of E at x are defined as respectively. It turns out that x → θ * (E, x) and x → θ * (E, x) are Borel functions that agree H n -a.e. on R n . Therefore, the n-dimensional density of E at x is defined for H n -a.e. x ∈ R n , and x → θ(E, x) is a Borel function on R n . Given t ∈ [0, 1], we set By the Lebesgue differentiation theorem, the pair (1) ) is called the essential boundary of E.

Rectifiable sets
where C 1 c (R n ) denotes the class of C 1 functions in R n with compact support. The relative perimeter of E in A ⊂ R n is then defined by setting P(E; A) := |μ E |(A) for any Borel set A ⊂ R n , and the perimeter of E is defined as P( exists and belongs to S n−1 .
If E is a set of locally finite perimeter, it turns out that

General facts about measurable functions
Let f : R n → R be a Lebesgue measurable function. We define the approximate upper limit f ∨ (x) and the approximate lower We observe that f ∨ and f ∧ are Borel functions that are defined at every point of Note that, by the above considerations, it follows that Let A ⊂ R n be a Lebesgue measurable set. We say that t ∈ R ∪ {±∞} is the approximate limit of f at x with respect to A, If this is the case, we say that ν f (x) := ν is the approximate jump direction of f at x. If we denote by J f the set of approximate jump points of f , we have that J f ⊂ S f and ν f : J f → S n−1 is a Borel function.

Functions of bounded variation
Let f : R n → R be a Lebesgue measurable function, and let ⊂ R n be open. We define the total variation of f in as is the set of C 1 functions from to R n with compact support. We also denote by C c ( ; R n ) the class of all continuous functions from to R n . Analogously, for any k ∈ N, the class of k times continuously differentiable functions from to R n is denoted by C k c ( ; R n ). We say that f belongs to the space of functions of bounded variations, In the special case can be written as Moreover, if f j = 0 (or, more in general, if f is a good representative, see [1, Theorem 3.28]), the total variation of D f can be obtained as where the supremum runs over all N ∈ N, and over all the possible partitions of (a, b) with a < x 1 < x 2 < · · · < x N < b. When n = 1, we will often write f instead of ∇ f .

Setting of the problem and preliminary results
In this section we give the notation for the chapter, and we introduce some results that will be extensively used later. For every x, y ∈ S n−1 , the geodesic distance between x and y is given by We recall that the geodesic distance satisfies the triangle inequality: for every x, y, z ∈ S n−1 .
Let r > 0, p ∈ S n−1 and β ∈ [0, π] be fixed. The open geodesic ball (or spherical cap) of centre r p and radius β is the set Note in the extreme cases β = 0 and β = π we have B 0 (r p) = ∅ and B π (r p) = ∂ B(r ) \ {−r p}, respectively. Accordingly, the geodesic sphere of centre r p and radius β is the boundary of B β (r p), which is given by The (n − 1)-dimensional Hausdorff measure of a geodesic ball and the (n − 2)-dimensional Hausdorff measure of a geodesic sphere are given by Let E ⊂ R n be a measurable set. For every r > 0, we define the spherical slice of radius r of E as the set Let v : (0, ∞) → [0, ∞) be a Lebesgue measurable function, and let E ⊂ R n be a measur- If E is spherically v-distributed, we can define the function is strictly increasing and smoothly invertible in (0, π).
Among all the spherically v-distributed sets of R n , we denote by F v the one whose spherical slices are open geodesic balls centred at the positive e 1 axis., i.e.
where α v is defined by (3.3)

Normal and tangential components of functions and measures
For every are the radial and tangential components of ϕ, respectively. (3.8) The following lemma gives some useful identities that will be needed later.
Recalling that ϕ = ϕ ⊥ + ϕ , combining (3.9) and (3.10) it follows that Proof First of all, note that where I represents the identity map in R n , andx ⊗x is the usual tensor product ofx with itself (so that I −x ⊗x is the orthogonal projection on the tangent plane to S n−1 atx). Thanks to (3.11), we have which proves (3.9). Note now that, by definition (3.8), it follows that On the other hand, from (3.9) Comparing last identity with (3.12) we obtain that for every Applying the last identity to the function ϕ we obtain (3.10).
If μ is an R n -valued Radon measure on R n 0 , we will write μ = μ ⊥ + μ , where μ ⊥ and μ are the R n -valued Radon measures on R n 0 such that for every ϕ ∈ C c (R n 0 ; R n ). Note that μ ⊥ and μ are well defined by Riesz Theorem (see, for instance, [1, Theorem 1.54]). In the special case μ = D f , with f ∈ BV loc (R n 0 ), we will shorten the notation writing D f and D ⊥ f in place of (D f ) and (D f ) ⊥ , respectively. In particular, if f = χ E and E ⊂ R n is a set of finite perimeter, by De Giorgi structure theorem we have Next lemma gives some useful identities concerning the radial and tangential components of the gradient of a BV loc function.
By definition of D f and thanks to (3.10) we have and this shows (3.14). Similarly, by definition of Thanks to (3.9), identity (3.15) follows.
An immediate consequence of identity (3.14) is the following.
We conclude this subsection with an important proposition, that is a special case of the Coarea Formula (see [ Proof The result follows by applying [1,Remark 2.94 In the next subsection we show how the notion of set of finite perimeter can be given in a natural way also for subsets of the sphere S n−1 (and, more in general, of ∂ B(r ), for any r > 0).

Sets of finite perimeter on S n−1
We now give a very brief introduction to sets of finite perimeter on S n−1 , by using the notion of integer multiplicity rectifiable currents, see [29,Chapter 6] for more details (see also [6]). Let k ∈ N with 1 ≤ k ≤ n − 1. We denote by k (R n ) and k (R n ) the linear spaces of k-vectors and k-covectors in R n , respectively, while D k (R n ) stands for the set of smooth k-forms with compact support in R n .
. . , τ k (x) an orthonormal basis for the approximate tangent space of M at x, and ·, · denotes the usual pairing between k (R n ) and k (R n ). In the special case when while the mass M(T ) of T is given by More in general, for any open set U ⊂ R n , we set Let A ⊂ S n−1 be an H n−1 -measurable set. We will say that A is a set of finite perimeter with the property that M U (Q) < ∞ for every U ⊂⊂ R n . By the Riesz representation theorem it follows that there exists a Radon measure μ Q and a μ Q -measurable function ν : for every smooth vector field with ϕ = ϕ . If A ⊂ S n−1 is a set of finite perimeter on the sphere, the reduced boundary ∂ * A is the set of points x ∈ S n−1 such that the limit exists, ν A (x) ∈ T x S n−1 , and ν A (x) = 1. The De Giorgi structure theorem holds true also for sets of finite perimeter on the sphere. In particular, ∂ * A is countably (n − 2)-rectifiable, for every smooth vector field with ϕ = ϕ . The isoperimetric inequality on the sphere states that, if β ∈ (0, π) and A ⊂ S n−1 is a set of finite perimeter on S n−1 with H n−1 (A) = H n−1 (B β (e 1 )), then (see [27]) The next theorem is a version of a result by Vol'pert (see [31]). (ii) for every r ∈ G E ∩ {0 < α v < π}: Proof The result follows applying [29,Theorem 28.5] with f (x) = |x|, and recalling the definition of slicing of a current (see [29,Definition 28.4]).
We now make some important remarks about Theorem 3.7.

Remark 3.8
Thanks to property (ib), we have Therefore, whenever r ∈ G E we will often write ∂ * E r instead of ∂ * (E r ) or (∂ * E) r , without any risk of ambiguity. Moreover, for every r ∈ G E we will also use the notation

Remark 3.9
In dimension n = 2, the theorem above implies that, if r ∈ G E ∩ {0 < θ < π}, Let now λ E be the measure defined in (1.5): If B ⊂ G E , then by (3.18) so that λ E (B) = 0. As a consequence, λ E is singular with respect to the Lebesgue measure in (0, ∞). If n > 2 this conclusion is in general false (unless one chooses E = F v , see Remark 3.10 below), and it may happen that λ E has a non trivial absolutely continuous part.

Remark 3.10
If n ≥ 2, but we consider the special case E = F v , Theorem 3.7 gives much more information than the one we can obtain for a generic set of finite perimeter. Indeed, let R ∈ O(n) be any orthogonal transformation that keeps fixed the e 1 axis.
Therefore, applying Theorem 3.7 to F v we infer that (j) for every r ∈ G F v : (jj) for every r ∈ G F v ∩ {0 < α v < π}: Therefore, Moreover, repeating the argument used in Remark 3.9 one obtains that Thus, the measure λ F v defined in (1.5) is purely singular with respect to the Lebesgue measure in (0, ∞).

Properties of v and v
In this section we discuss several properties of the functions v and ξ v . These are the natural counterpart in the spherical setting of analogous results proven in [2,11]. We start by showing that, if E ⊂ R n is a set of finite perimeter and volume, then v ∈ BV (0, ∞).

Lemma 4.1
Let v be as in Theorem 1.1, and let E ⊂ R n be a spherically v-distributed set of finite perimeter and finite volume.
Proof We divide the proof into steps. Step Let now ψ ∈ C 1 c (0, ∞) with |ψ| ≤ 1. Applying formula (3.9) to the radial function ψ(|x|)x, we obtain that for every By Coarea formula, the integral in the left hand side can be written as (4.5) Combining (4.4) and (4.5) we find that Taking the supremum over ψ we obtain that Step 2 We conclude the proof. Since the function r → 1/(r n−1 ) is smooth and locally bounded in (0, ∞), we also have that ξ v (r ) ∈ BV loc (0, ∞). Moreover, recalling that v(r ) = r n−1 ξ v (r ), by the chain rule in BV (see [1,Example 3.97]) Let now ψ ∈ C 1 c (0, ∞). From the previous identity it follows that Combining the previous identity and (4.6), By approximation, the identity above is true also when ψ is a bounded Borel function, and this gives (4.1).
Taking the supremum over all such ψ gives By approximation, the inequality above holds true for every Borel set, and this shows inequality (4.2).
The next lemma gives an important property of the measure r n−1 Dξ v .

Lemma 4.2
Let v be as in Theorem 1.1, and let E ⊂ R n be a spherically v-distributed set of finite perimeter and finite volume. Then for every x ∈ S α v (r ) (re 1 ).
We now prove an auxiliary inequality that will be useful later.

Proposition 4.3 Let v be as in
(4.10) In the following, it will be convenient to introduce the function V j : (0, ∞) → R given by where α v j : (0, r ) → [0, π] is defined by (3.6), with v j in place of v. We divide the proof into several steps.
Step 1 We show that V j is Lipschitz continuous with compact support. Indeed, Moreover, for every r 1 , r 2 ∈ (0, ∞), where we used the fact that ξ v j is compactly supported in (0, ∞) (since v j is), and r 1 and r 2 are such that Step 2 We show that α v j is H 1 -a.e. differentiable and that for H 1 -a.e. r > 0. Let us set ϕ(r ω) · ω dH n−2 (ω) This shows (4.11) whenever r ∈ A j . Note now that where Int(·) stands for the interior of a set. Since α v j (r ) = 0 for every r ∈ Int({α v j = 0}) ∪ Int({α v j = π}), using the identities above one can see that (4.11) holds true for H 1 -a.e. r > 0.
Integrating (4.11), thanks to the classical divergence theorem applied in , and recalling that V j has compact support, we obtain which gives the claim.
Step 4 we prove that where (supp ϕ) ⊂ (0, ∞) is the compact set defined in Step 1. Thanks to (4.10) and Step 3 We now estimate the right hand side of the expression above. Thanks to (3.6) and arguing as in Step 2 we have that r n−1 ξ v j (r ) dr = r n−1 Dξ v j ( (supp ϕ)). (4.14) Let us now focus on the second integral in the right hand side of (4.13). Applying the divergence theorem (3.16) with A = B α v j (r ) (re 1 ), and denoting by ν * (x) the exterior unit normal to S α v j (r ) (re 1 ), we have Combining (4.13), (4.14), and (4.15), we obtain (4.12).
Step 5 We show that F v is a set of finite perimeter. Note that and α v j → α H 1 -a.e. in (0, ∞). Note also that, from our choice of the sequence {v j } j∈N and thanks to (4.7), it follows that Therefore, taking the limsup as j → ∞ in (4.12), and using the fact that (supp ϕ) is compact, where we also used the isoperimetric inequality in the sphere (see (3.17)) and the Coarea formula. Taking the supremum of the above inequality over all functions ϕ ∈ C 1 c ( ( × S n−1 ); R n ) with ϕ L ∞ ( ( ×S n−1 );R n ) ≤ 1, we obtain P(F v ; ( × S n−1 )) ≤ r n−1 Dξ v ( ) + P(E; ( × S n−1 )).
Thanks to (4.2) we have since E is a set of finite perimeter by assumption. Since was arbitrary, this shows that F v is a set of locally finite perimeter.
Taking the supremum over all for any open set ⊂ (0, ∞) with B ⊂ . Taking the infimum of the above inequality over all open sets ⊂ (0, ∞) with B ⊂ , we obtain inequality (4.9) when B is a Borel set.

Proof of Theorem 1.1
In this section we prove Theorem 1.1, and state some important auxiliary results. The proof of Lemma 1.3 is postponed to Sect. 6, since it requires some results related to the circular symmetrisation. We start by proving Theorem 1.1.
Proof of Theorem 1. 1 We will adapt the arguments of the proof of [2, Theorem 1.1]. Let G F v be the set associated with F v given by Theorem 3.7. We start by proving (1.4). We will first prove the inequality when B ⊂ (0, ∞) \ G F v , and then in the case B ⊂ G F v . The case of a general Borel set B ⊂ (0, ∞) then follows by decomposing B as Step 1 We prove inequality (1.4) when B ⊂ (0, ∞) \ G F v . First observe that, thanks to Proposition 3.6 and (3.13), where in the last inequality we used (4.2).
Step 2 We prove inequality (1.4) when B ⊂ G F v . We divide this part of the proof into further substeps.
Step 2b We show that We will now use the fact that, by duality, we can write where {w h } h∈N is a countable dense set in (−1, 1). Then, thanks to (5.5) where we applied identity (5.6) with t = g(r )/ p E (r ), and we also used the fact that p E (r ) = 0 for H 1 -a.e. r / ∈ {0 < α v < π}, thanks to Volper't theorem. Applying Lemma 3.1 to the functions we obtain (5.4).
Step 2c We conclude the proof of Step 2. In the special case E = F v , thanks to Vol'pert Theorem and Lemma 4.2 we have Using the isoperimetric inequality (3.17) together with (5.4) and (5.3) we then have, from which we conclude.
Step 3 We conclude the proof of the theorem. Suppose P(E) = P(F v ). Then, in particular, all the inequalities in Step 2 hold true as equalities. At the end of Step 2c we used the fact that, by the isoperimetric inequality (3.17), we have If the above becomes an equality, this means that for H 1 -a.e. r ∈ {0 < α v < π} the slice E r is a spherical cap. Finally, the fact that for H 1 -a.e. r ∈ {0 < α v < π} we have follows from Vol'pert Theorem 3.7, and this shows (a). Let us now prove (b). If P(E) = P(F v ), the Jensen's inequality at the end of Step 2b, for the strictly convex function becomes an equality. This implies that for H 1 -a.e. r ∈ {0 < α v < π} the function is H n−2 -a.e. constant in ∂ * E r . Therefore, the two functions The previous result allows us to prove a useful proposition (see also [2,Proposition 3.4]).

2) it follows that
where we also used the fact that p E = 0 H 1 -a.e. in B, since Let us now assume B ⊂ G F v . In this case, by Lemma 4.2 we have |r n−1 D s ξ v |(B) = 0. Then, thanks to (5.3) and (5.4) we obtain so that (5.8) follows. Consider now the case E = F v . If B ⊂ G F v , recalling again that by Lemma 4.2 we have |r n−1 D s ξ v |(B) = 0, thanks to (5.7) we obtain Therefore, thanks to (5.2), An important consequence of the above proposition is a formula for the perimeter of F v .

Corollary 5.2
Let v : (0, ∞) → [0, ∞) be a measurable function satisfying (1.2) such that F v is a set of finite perimeter and finite volume. Then We conclude this section with an important result, that will be used later.

Remark 5.4
Note that the equivalence (iii) ⇐⇒ (i) holds true also if I is a Borel set. To show this, we only need to prove that (i) ⇒ (iii), since the opposite implication is given by repeating Step 3 of the proof of Proposition 5.3. Suppose (i) is satisfied. Then from (4.8) we have r n−1 Dξ v I = r n−1 ξ v I . Therefore, thanks to (5.9) Proof We divide the proof into three steps.
Step 3 (iii) ⇒ (i) (note that we will not use the fact that I is open). Assume (iii) holds true. Then, where we used the fact that H 1 (B 0 ) = 0, thanks to (3.19).

Circular symmetrisation and proof of Lemma 1.3
In this section we show Theorem 1.4, Lemma 1.5, and finally Lemma 1.3. We will only sketch the proofs, since in most cases the arguments follow the lines of the proofs in Sects. 3, 4, and 5. We start with some notation which, together with that one already given in the Introduction, will be extensively used in this section. Let (r , x ) ∈ (0, ∞) × R n−2 , β ∈ [0, π], and let p ∈ S 1 . The circular arc of centre (r p, x ) and radius β is the set Note that in this case the relation between α and ξ is linear. If μ is an R n -valued Radon measure on R n \ {x 12 = 0}, we will write μ = μ 12⊥ + μ 12 , where μ 12⊥ and μ 12 are the R n -valued Radon measures on R n \ {x 12 = 0} such that for every ϕ ∈ C c (R n \ {x 12 = 0}; R n ). The next two results play the role of Proposition 3.6 and Vol'pert Theorem 3.7, in the context of circular symmetrisation.
for every Borel set B ⊂ (0, ∞) × R n−2 , and where ∇ x denotes the approximate gradient of with respect to x .
The next result should be compared to Proposition 4.3.
We are now ready to prove Theorem 1.4.

Proof of Theorem 1.4
Using the results shown above, Theorem 1.4 can be proved by following the lines of the proof of Theorem 1.1.
We will now state the results that are needed to prove Lemma 1.5. The next proposition should be compared to Proposition 5.1.
Moreover, in the special case E = F , equality holds true.
A straightforward consequence of the previous result is the following formula for the perimeter of F .
loc ( ) and ∈ W 1,1 loc ( ); We can now prove Lemma 1.3. As already mentioned in the Introduction, the proof relies on Theorem 1.4 and Lemma 1.5.
Proof of Lemma 1. 3 We divide the proof into steps.
To this aim, we first prove an auxiliary result.
Step 2a We show that if F ⊂ R n is a set of finite perimeter such that (F) r is a spherical cap for H 1 -a.e. r > 0, and Here, the vector ν F 1 j is defined in the following way. Let j ∈ {2, . . . , n}, and let ν F 1 j be the orthogonal projection of ν F on the bi-dimensional plane generated by e 1 and e j . In this plane, we consider the following orthonormal basis { x 1 j , x 1 j }: where x 1 j is directed along the radial direction, and x 1 j is parallel to the tangential direction.
To show the claim, first of all note that, by Vol'pert Theorem 3.7, for H 1 -a.e. r > 0 we have Then, thanks to (6.1), Step 2b We conclude. Let E 1 := E, and let E 2 be set obtained by applying to E the circular symmetrisation with respect to (e 1 , e 2 ). Then, for j = 3, . . . , n, we define iteratively the set E j as the circular symmetral of E j−1 with respect to (e 1 , e j ). Note that, since H 1 -a.e. spherical section of E is a spherical cap, we have E n = F v . Therefore, thanks to the perimeter inequality (1.11) under circular symmetrisation (see Theorem 1.4), we have Moreover, for j = 3, . . . , n, we define I j := (I × S n−1 ) ∩ {x j = 0} ∩ {x 1 > 0}. It is not difficult to check that Then, applying Lemma 1.5 to F v and E n−1 , we obtain that which, in turns, implies Applying iteratively this argument to E n−2 , . . . , E, we conclude.

Proof of Theorem 1.2: (ii) ⇒ (i)
Before giving the proof of the implication (ii) ⇒ (i) of Theorem 1.2, it will be convenient to introduce some useful notation. Let v and I = {0 < α ∧ v ≤ α ∨ v < π} be as in the statement of Theorem 1.2. By assumption, I is an interval and α v ∈ W 1,1 loc (I ) where, to ease the notation, we set I :=I. Let now E be a spherically v-distributed set of finite perimeter. We define the average direction of E as the map d E : I → S n−1 given by where G E ⊂ (0, ∞) is the set given by Theorem 3.7. To ease our calculations, it will also be convenient to introduce the barycentre function b E : The importance of the functions d E and b E is given by the following lemma.
Proof Let us immediately observe that (7.2) follows by the definitions of d E and b E . By assumption, for H 1 -a.e. r ∈ I , there exists ω(r ) ∈ S n−1 such that E r = B α v (r ) (r ω(r )). We are left to show that Note that for H 1 -a.e. r ∈ I we have E r = B α v (r ) (r ω(r )) and ∂ * E r = S α v (r ) (r ω(r )).

Remark 7.2
Let us point out that here we are using the term barycentre in a slightly imprecise way. Indeed, for a given r ∈ I ∩ G E , the geometric barycentre of E r is given by Nevertheless, we will still keep this terminology, since b E turns out to be very useful for our analysis.
We are now ready to prove the implication (ii) ⇒ (i) of Theorem 1.2.

Proof of Theorem 1.2: (ii) ⇒ (i). Suppose (ii) is satisfied, and let E ∈ N (v).
We are going to show that there exists an orthogonal transformation R ∈ SO(n) such that H n (E (R F v )) = 0. We now divide the proof into steps.
Step 1 First of all, we observe that Indeed, since α v ∈ W 1,1 loc (I ), thanks to Proposition 5.3 we have Since E ∈ N (v), applying Lemma 1.3 the claim follows.
Step 2 We show that b E ∈ W 1,1 loc (I ; R n ) and Indeed, let ψ ∈ C 1 c (I ) be arbitrary, and let i ∈ {1, . . . , n}. By definition of b E Note now that div x i |x| n ψ(|x|)x = x i |x| n ψ (|x|).
Step 3 We show that for H 1 -a.e. r ∈ I . (7.7) Since E ∈ N (v), from Theorem 1.1 we know that for H 1 -a.e. r ∈ I the spherical slice E r is a spherical cap. Then, thanks to Lemma 7.1 for H 1 -a.e. r ∈ I .
Still thanks to Theorem 1.1, we know that for H 1 -a.e. r ∈ I the functions for some measurable functions a : I → (−1, 1) and c : I → (0, 1]. Therefore, recalling the definition of d E together with (7.4)-(7.5) we obtain Note now that from Step 1 and (4.8) it follows that for H 1 -a.e. r ∈ I Plugging last identity into (7.8) and using (7.2), we obtain where we used the fact that, thanks to (3.1) and (3.3), for H 1 -a.e. r ∈ I .
Step 4 We conclude. First of all, note that from (7.2) and Step 2 it follows that d E ∈ W 1,1 loc (I ; S n−1 ). Then, thanks to Step 3, for H 1 -a.e. r ∈ I cos α v (r ) (sin α v (r )) n b E (r ) + b E (r ) − n − 1 (sin α v (r )) n (cos α v (r ))α v (r ) = 0, for H 1 -a.e. r ∈ I . This shows that d E is H 1 -a.e. constant in I . Therefore, E ∩ (I × S n−1 ) can be obtained by applying an orthogonal transformation to F v ∩ (I × S n−1 ).

Proof of Theorem 1.2: (i) ⇒ (ii)
We start by showing that the fact that {0 < α ∧ ≤ α ∨ < π} is an interval is a necessary condition for rigidity.

Proposition 8.1
Let v : (0, ∞) → [0, ∞) be a measurable function satisfying (1.2), such that F v is a set of finite perimeter and finite volume, and let α v be defined by (1.3). Suppose that the set {0 < α ∧ ≤ α ∨ < π} is not an interval. That is, suppose that there exists r ∈ {α ∧ = 0} ∪ {α ∨ = π} such that Then, rigidity fails. More precisely, setting Before giving the proof of Proposition 8.1 we need the following lemma.
We divide the proof in two steps.
Step 1 We show that . To this aim, it will be enough to show that Let us first prove that For this reason, we will assume α ∨ v (r ) < π. Note now that (8.2) follows if we prove that Let now x ∈ ∂ B(r ), and suppose that there exists δ > 0 such that Let now ρ > 0 be so small that dist S n−1 (ŷ,x) < δ 2 for every y ∈ B(x, ρ).
By triangle inequality for the geodesic distance we have, in particular, that Since, by Step 1, We can now give the proof of Proposition 8.1.

Proof of Proposition 8.1 Note that, since B(r ) is open and E
From this, it follows that Similarly, we obtain Thus, thanks to (8.5) and (8.6) Therefore, in order to conclude the proof we only need to show that Without any loss of generality, we will assume that Let now E 1 , E 2 , and R be as in the statement. We divide the proof of (8.7) into steps.
Step 1 We show that To this aim, it will be enough to prove that If α ∨ v (r ) = π inequality (8.9) is obvious, so we will assume that α ∨ v (r ) < π.
Step 1a We show that 1 . Indeed, let x ∈ ∂ B(r ), and suppose that there exists δ > 0 such that Step 1c We conclude the proof of Step 1. By definition of E, from Step 1a and Step 1b it follows that Step 2 We show (8.7), concluding the proof. Thanks to Step 1 and Lemma 8.2 we have where we also used (1.4) with B = {r }.
We now show that, if the jump part D j α v of Dα v is non zero, rigidity fails.

Proposition 8.3
Let v : (0, ∞) → [0, ∞) be a measurable function satisfying (1.2) such that F v is a set of finite perimeter and finite volume, and let α v be defined by (1.3). Suppose that α v has a jump at some point r > 0. Then, rigidity fails. More precisely, setting Proof Let R ∈ O(n), λ ∈ (0, 1), and E ∈ R n be as in the statement, and set ω := Re 1 . Arguing as in the proof of Proposition 8.1 we have: Therefore, in order to conclude the proof we only need to show that Without any loss of generality, we will assume that We now proceed by steps.
Step 1 We show that To show (8.13), it is enough to prove that for every x ∈ (∂ * E) r we have for every x ∈ (∂ * E) r , (8.14) and which shows (8.17). This, together with (8.16), implies (8.14). As already mentioned, (8.15) can be proved in a similar way, and therefore (8.13) follows.
Step 2 We conclude. From (8.10) it follows that Therefore, thanks to (8.13) and Lemma 8.2 where we also used (1.4) with B = {r }. Then, (8.11) follows from the last chain of inequalities.
We conclude this section showing that, if D c α v = 0, rigidity fails.

Proposition 8.4
Let v : (0, ∞) → [0, ∞) be a measurable function satisfying (1.2) such that F v is a set of finite perimeter and finite volume, and let α v be defined by (1.3). Suppose that D c α v = 0. Then, rigidity fails.
Proof We are going to construct a spherically v-distributed set E ∈ N (v) that cannot be obtained by applying a single orthogonal transformation to F v (see (8.20)

below).
First of all, let us note that it is not restrictive to assume that α v is purely Cantorian. Indeed, by (2.4) one can decompose α v into where α a v ∈ W 1,1 loc (0, ∞), α j v is a purely jump function, and α c v is purely Cantorian. Thanks to (8.18), in the general case when α v = α c v , the proof can be repeated by applying our argument just to the Cantorian part α c v of α v . Therefore, from now on we will assume that Thanks to Proposition 8.1, we can also assume that {0 < α ∧ v ≤ α ∨ v < π} is an interval (otherwise there is nothing to prove, since rigidity fails). Moreover, since α v is continuous, there exist a, b > 0, with a < b, such that I := (a, b) ⊂⊂ {0 < α ∧ v ≤ α ∨ v < π} and 0 < α v (r ) < π for every r ∈ I . (8.19) Since D c α v = 0, it is not restrictive to assume |D c α v |(I ) > 0. For each γ ∈ (−π, π), we define R γ ∈ O(n) in the following way: x 1 cos γ − x 2 sin γ x 1 sin γ + x 2 cos γ x 3 . . .
First of all note that, by (3.5) and by the chain rule in BV (see, [1,Theorem 3.96]), it follows that ξ v is purely Cantorian, where ξ v is given by (3.3). Moreover, from (2.5) and from the fact that ξ v is continuous, we have where the supremum runs over N ∈ N and over all r 1 , . . . , r N with a < r 1 < r 2 < · · · < r N < b. Therefore, for every k ∈ N there exist N k ∈ N and r k 1 , . . . , r k N with a < r k 1 < r k 2 < · · · < r k N < b such that Without any loss of generality, we can assume that the partitions are increasing in k. That is, we will assume that r k 1 , . . . , r k N k ⊂ r k+1 1 , . . . , r k+1 N k+1 for every k ∈ N.
Define now, for every k ∈ N, where we set r k 0 := a and r k N k +1 := b. Let us now set v k (r ) := ξ k v (r )/r n−1 for every r ∈ I and for every k ∈ N, using (8.24) and the fact that ξ v is continuous we obtain where we also used (8.23).
Step 2 For each k ∈ N, we construct a spherically v k -distributed set E k such that P(E k ; (I × S n−1 )) = P(F v k ; (I × S n−1 )).
From ( where we also used (1.4).
Step 4 We conclude. Let E be given by (8.20). Then, E is spherically v-distributed and satisfies where E is defined in (8.28). By repeating the arguments used in the proof of Proposition 8. Acknowledgements The authors would like to thank Marco Cicalese, Nicola Fusco, and Emanuele Spadaro for inspiring discussions on the subject. They would also like to thank Frank Morgan for useful comments on a preliminary version of the paper. F. Cagnetti was supported by the EPSRC under the Grant EP/P007287/1 "Symmetry of Minimisers in Calculus of Variations".
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