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The Besicovitch covering property in the Heisenberg group revisited

The Besicovitch covering property (BCP) is known to be one of the fundamental tools in measure theory, and more generally, a usefull property for numerous purposes in analysis and geometry. We prove both sufficient and necessary criteria for the validity of BCP in the first Heisenberg group equipped with a homogeneous distance. Beyond recovering all previously known results about the validity or non validity of BCP in this setting, we get simple descriptions of new large classes of homogeneous distances satisfying BCP. We also obtain a full characterization of rotationally invariant distances for which BCP holds in the first Heisenberg group under mild regularity assumptions about their unit sphere.


Introduction
The theory of differentiation of measures originates from works of Besicovitch in the Euclidean space ( [1], [2]). These pioneering works, as well as subsequent developments of the theory in the metric setting, rely as fundamental tools on suitable covering properties of the ambient space. Among such covering properties, the Besicovitch covering property (BCP), see Definition 1.2, plays a central role for the validity of the differentiation theorem for every locally finite Borel regular measure over a separable metric space, see for instance [7] or the survey [8]. We also stress that, beyond its role in the theory of differentiation of measures, BCP is a covering property that is geometric in nature and can be used as a useful tool to deduce global properties on the ambient space from local ones.
We focus in the present paper on the first Heisenberg group H with the aim of describing new homogeneous distances satisfying BCP. Our results come in the continuity of [6] where the existence of homogeneous distances for which BCP holds on H was first proved. We prove in the current paper sufficient conditions, see Theorem 1.4 and Section 3, and necessary conditions, see Theorem 1.5 and Section 4, for the validity of BCP on the first Heisenberg group equipped with a homogeneous distance. Beyond recovering the class of homogeneous distances for which it was proved in [6] that BCP holds, we will get as a consequence of our sufficient conditions rather simple descriptions of new large classes of homogeneous distances on H for which BCP holds, see Section 3.1. We stress that the proof of Theorem 1.4 requires original arguments that are independent from those used in [5] or [6]. As a consequence of Theorem 1.4 and Theorem 1.5, we will also get a simple characterization of rotationally invariant homogeneous distances on H for which BCP holds, under mild additional assumptions about the regularity of their unit sphere, see Theorem 1.6. One can for instance easily recover from these results the fact, first proved in [9], that the Carnot-Carathéodory distance on H does not satisfy BCP. One also recovers the fact that the Korányi distance do not satisfy BCP, this was already noticed in [3] and [10]. More importantly, one gets explicit homogeneous distances that are as close as one wants to the Korányi distance and that do satisfy BCP, see Theorem 1.9.
Before stating our main results, we recall the definition of the Besicovitch covering property in the metric setting. Here, balls in a metric space (X, d) are assumed to be closed. Namely, a ball denotes a set of the form B(p, r) := {q ∈ X : d(p, q) ≤ r} for some p ∈ X and some 0 < r < ∞. In this paper, we identify the first Heisenberg group H with R 2 × R equipped with the group law given by , v = (v 1 , v 2 ) ∈ R 2 , and equipped with the family of dilations (δ t ) t≥0 defined by δ t (v, z) := (tv, t 2 z). We recall that a distance d on H is said to be homogeneous if it is left-invariant and one-homogeneous with respect to the family of dilations, see Section 2.
For the sake of completeness, we recall that the validity of BCP for a homogeneous distance d on H characterizes the validity of the differentiation theorem for every Radon measure over (H, d). Theorem 1.3 below is indeed a corollary of [5,Theorem 1.5]. We refer to [7] or [8] for more general statements in the metric setting. f (y) dλ(y) = f (x) for λ-a.e. x ∈ X for every f ∈ L 1 loc (λ), if and only if (H, d) satisfies BCP. We briefly describe now our main results. Given a homogeneous distance d on H, we know from [4] that its unit ball centered at the origin B can be described as for some compact set K ⊂ R 2 and some function φ : K → [0, +∞) locally Lipschitz on int(K), which we call the profile of the homogeneous distance d. We refer to Section 2 for more details about profiles of homogeneous distances.
The sufficient and necessary conditions for the validity of BCP proved in the present paper will be mainly given in terms of suitable properties for the profile of homogeneous distances.
Namely, we prove the following sufficient condition, Theorem 1.4 below.
Theorem 1.4 will be obtained through an application of an abstract sufficient condition proved in Section 3, see Theorem 3.12 and its corollary, Corollary 3.15.
Conversely, we also prove several necessary conditions. For homogeneous distances, without any further assumption on the regularity of their profile, our main necessary conditions for the validity of BCP are gathered in Theorem 1.6 below. We refer to Section 4 for more detailed statements. We denote by S 1 the Euclidean unit circle in R 2 . Theorem 1.5. Let d be a homogeneous distance on H with profile φ : K → [0, +∞). Assume that BCP holds in (H, d). Then there is a countable closed set W ⊂ S 1 such that S 1 \ W can be written as a countable union of relatively open subsets O j , and, for each j, there are m j > 0 and α j ∈ (0, π/2) such that, for all v ∈ int(K)\{0} such that φ is differentiable at v and v/ v ∈ O j , we have, for all w ∈ R 2 \ {0}, Combining Theorem 1.4 and Theorem 1.5, one gets a particularly simple characterization, Theorem 1.6 below, for the validity of BCP for rotationally invariant homogeneous distances, under mild regularity assumptions on their profile. Namely, we say that a homogeneous distance d on H is rotationally invariant if rotations around the z-axis -in the Euclidean senseare isometries in (H, d). Equivalently, a homogeneous distance d on H with unit ball centered at the origin B is rotationally invariant if and only if the intersection of B with the v-plane is a Euclidean disc with radius r d for some r d > 0 and B can be described as which we call the radial profile of the rotationally invariant homogeneous distance d. Theorem 1.6. Let d be a rotationally invariant homogeneous distance on H with radial profile ϕ : [−r d , r d ] → [0, +∞). Assume that |ϕ (s)/s| is essentially bounded in a neighbourhood of 0, which holds in particular if ϕ ∈ C 1 ((−r d , r d )) with ϕ is differentiable at 0. Then BCP holds in (H, d) if and only if there is m > 0 such that ϕ (s) ≤ −ms for a.e. s ∈ (0, r d ).
Theorem 1.9 below gives an application of this characterization. We recall that the Korányi distance d 0 is the homogeneous distance on H given by For > 0, we define d as the homogeneous distance on H given by Note that these homogeneous distances are rotationally invariant. Note also that As already said, it has already been noticed, independently and at the same time, in [3] and [10], that BCP does not hold for the Korányi distance. On the other hand, it has been proved in [6] that BCP holds in (H, d 1 ). As an application of Theorem 1.6, we can hence recover both results, but, more importantly, we get the validity of BCP for all homogeneous distances d with > 0. This gives in particular an example of a metric space (H, d 0 ) that does not satisfy BCP but whose distance d 0 is as close as one wants from bi-Lipschitz equivalent distances for which BCP holds.
Going back to the necessary conditions and to homogeneous distances that are not assumed to be rotationally invariant, we will prove some improvements of Theorem 1.5 under further regularity assumptions on the profile of the homogeneous distance, see Section 4.3. In particular, Theorem 1.10 below is a consequence of the results proved in this last section of the paper. Theorem 1.10. Let d be a homogeneous distance on H with profile φ : K → [0, +∞). Assume that BCP holds in (H, d) and assume that φ ∈ C 2 (int(K)). Then ∇φ(w), w < 0 for all w ∈ int(K) \ {0}, φ(w) < φ(0) for all w ∈ K \ {0}, and the Hessian of φ at 0, Hφ(0), is semi-definite negative. Moreover, if Hφ is assumed to be differentiable at 0, then Hφ(0) is definite negative.
To conclude this introduction, we stress that it seems to be a quite delicate issue to fill in the little gap between the sufficient and the necessary conditions proved in the present paper in order to get a characterization of non-rotationally invariant homogeneous distances satisfying BCP on H, even under the additional assumptions about the regularity of their profile. However, our results should be strong enough to cover the study of the validity of BCP for most explicit homogeneous distances one may work with in H.
The rest of the paper is organized as follows. In Section 2, we recall our conventions about the Heisenberg group H, and we refine the description proved in [4] about unit balls centered at the origin for homogeneous distances on H. We also fix some conventions and notations for later use. Applications of Theorem 1.4 are given in Section 3.1, see for instance Corollary 3.8 and Corollary 3.9, which give the "if" part in Theorem 1.6, and the application for the proof of the validity of BCP for the homogeneous distances d for > 0. We prove in Section 3.2 an abstract sufficient condition on which the proof of Theorem 1.4 will be based. The proof of Theorem 1.4 itself is given in Section 3.3. Section 4 is devoted to the proof of several necessary conditions for the validity of BCP. Our arguments to get these necessary conditions are based on Proposition 4.3 which is a consequence of a sufficient condition for the non-validity of BCP proved in [6], see Section 4.1. The proof of Theorem 1.5 can be found in Section 4.2. We also refer to Theorem 4.10, Theorem 4.16 and Theorem 4.17 for more detailed statements. The "only if" part in Theorem 1.6 will be given by Corollary 4.14 and, as one of its applications, we will recover the fact that BCP does not hold for the Korányi distance. In section 4.3, we show several improvements of the necessary conditions proved in the previous section under additional regularity assumptions about the profile of homogeneous distances. Theorem 1.10 follows in particular from these improvements.

Preliminaries
We recall from the introduction that we identify the first Heisenberg group H with R 2 × R equipped with the group law given by where The identity element coincides with the origin denoted by 0 = (0, 0) and, for p ∈ H, we have p −1 = −p.
We equip H with the family of dilations (δ t ) t≥0 given by This family of dilations is a one parameter group of group automorphisms in H.
Definition 2.1 (Homogeneous distances). A distance d on H is said to be homogeneous if it is left-invariant, that is, d(p · q, p · q ) = d(q, q ) for all p, q, q ∈ H, and one-homogeneous with respect to the family of dilations (δ t ) t>0 , that is, d(δ t (p), δ t (q)) = td(p, q) for all t > 0 and all p, q ∈ H.
We recall that a homogeneous distance induces the Euclidean topology on H, see [5,Proposition 2.26]. Given a homogeneous distance d on H, we denote by B := {p ∈ H : d(0, p) ≤ 1} its unit ball centered at the origin. We define π : H → R 2 by π(v, z) := v.
The following description of unit balls centered at the origin for homogeneous distances on H will play a central role in the present paper.
Proposition 2.2. Let d be a homogeneous distance on H and let K := π(B). Then K is a compact convex set such that 0 ∈ int(K) and K is symmetric, that is, K = −K, and there is a bounded function φ : K → [0, +∞) locally Lipschitz on int(K) such that Furthermore, and We call the function φ : K → [0, +∞) given by Proposition 2.2 the profile of the homogeneous distance d.
The first part of Proposition 2.2 follows from [4, Proposition 6.2]. To prove (2.3), we first note that, since 0 ∈ int(B), there is z > 0 such that (0, z) ∈ B and hence φ(0) ≥ z > 0. Next, let v ∈ int(K) \ {0}. Assume with no loss of generality that v = (x, 0) with x > 0. Since v ∈ int(K), there is h > 0 such that v h := (x, h), v −h := (x, −h) ∈ int(K), and hence (v h , 0), (v −h , 0) ∈ B. By left-invariance and homogeneity of the distance, it follows that for all t ∈ [0, 1] by homogeneity of the distance. This implies that t 2 φ(v) ≤ φ(tv), and hence φ(v) ≤ lim inf t→1 − φ(tv). Conversely, since B is closed, φ is upper semicontinuous on K. In particular, it follows that, Note that a compact convex and symmetric set K ⊂ R 2 contains 0 in its interior if and only if K is non empty with K = int(K). Furthermore, note that Proposition 2.2 implies that Next, note that if d is a rotationally invariant homogeneous distance with radial profile ϕ : [−r d , r d ] → [0, +∞) (see the paragraph before Theorem 1.6) then K = π(B) is a closed Euclidean disc with radius r d centered at the origin and, if φ : K → [0, +∞) denotes its profile, we have φ(v) = ϕ( v ) for every v ∈ K. Hence (2.4) together with the fact that ϕ ∈ Lip loc ((−r d , r d )) imply that the radial profile is continuous on the closed segment [−r d , r d ], and hence its profile φ is continous on the closed set K.
However, we stress that the profile φ : K → [0, +∞) of a non-rotationally invariant homogeneous distance d on H, although continuous on int(K) and satisfying (2.4), need not be continuous on K as shows Example 2.7 below. We first give a criterion that might have its own interest and gives a large class of examples of profiles of homogeneous distances on H.
Proposition 2.5. Let K ⊂ R 2 be a compact convex and symmetric set containing 0 in its interior. Let φ : K → [diam(K) 2 /16, +∞) and assume that φ is concave on int(K) and upper semicontinuous on K. Then φ is the profile of a homogeneous distance on H. Namely, setting defines a homogeneous distance on H whose unit ball centered at the origin is the set B.
Proof. To prove that such a function is the profile of a homogeneous distance on H, we need to prove that the set B is a compact symmetric set that contains 0 in its interior and that for all p, q ∈ B and all t ∈ [0, 1] (see [5,Example 2.37]). By positivity and upper semicontinuity of φ on the compact set K, the set B is bounded and closed, hence compact. The facts that B is symmetric, that is, p −1 = −p ∈ B for every p ∈ B, and contains 0 in its interior are straightforward.
To prove (2.6), we note that B is the closure of the set Hence it is sufficient to prove that δ t (p) · δ 1−t (q) ∈ B + for every p, q ∈ B + and t ∈ [0, 1], together with the similar statement replacing B + by B − . We consider the case p, q ∈ B + , the other case being similar. Let p = (v, z) ∈ B + and q = (v,z) ∈ B + . Then Next, by definition of B + and since v , v ≤ diam(K)/2, we have Now, we conclude the proof with the following observation. For Since the function g : [0, 1] → R given by g(t) : 1]. Applying this observation with A = 4φ(v)/ diam(K) 2 and B = 4φ(v)/ diam(K) 2 and using the concavity of φ completes the proof.
We give in the next example an application of Proposition 2.5 that gives a profile φ : K → [0, +∞) of a homogeneous distance on H that is not continuous on K.
Then φ 1 : D → [1/4, +∞) is the profile of a homogeneous distance on H but φ 1 is not continuous on D.
is not continuous at (−1, 0) and (1, 0). The fact that φ 1 is the profile of a homogeneous distance will now follow from Proposition 2.5 if we show that φ 1 is concave on int(D). To this aim, we set and we prove that f is convex on int(D). We have Hence the determinant and the trace of the Hessian H f of f are given by that are both non negative on int(D). Hence f is convex on int(D) as claimed.
Note that it will follow from the necessary conditions proved in Section 4 that the homogeneous distance with profile φ 1 does not satisfy BCP, see Remark 4.24. We refer to Example 3.4 for another example of a homogeneous distance with non-continuous profile that satisfy BCP.
We end this section with conventions and notations that will be used in the rest of the paper. We denote by the same notations, namely, ·, · and · , the Euclidean scalar product and Euclidean norm on R 2 or R 3 . For n = 2, 3, and v, w ∈ R n \ {0}, we denote by (v, w) the unoriented angle between v and w given by Given v, w ∈ R 2 \ {0}, the oriented angle ∠(v, w) ∈ (−π, π) between v and w is well defined whenever w / ∈ R <0 v and is given by We also recall that, given any two vectors . We denote by S 1 the Euclidean unit circle in R 2 .

Sufficient conditions
This section is devoted to the proof and applications of Theorem 1.4, restated below for the reader convenience. We refer to Proposition 2.2 for the definition and general properties of the profile of a homogeneous distance on H.
Then BCP holds on (H, d).
We first give some applications of Theorem 3.1 in Section 3.1. We prove in Section 3.2 an abstract sufficient condition, Theorem 3.12, for the validity of BCP on H, or more generally on homogeneous groups, see Remark 3.14, equipped with a homogeneous distance. The proof of Theorem 3.1, or equivalently of Theorem 1.4, will then be given in Section 3.3 and will be based on a corollary of this abstract sufficient condition, Corollary 3.15.
3.1. Applications and Examples. First, combining Proposition 2.5 and Theorem 3.1 gives the following class of homogeneous distances that satisfy BCP on H. Namely, if K ⊂ R 2 is a compact convex and symmetric set containing 0 in its interior and φ : K → [diam(K) 2 /16, +∞) is concave on int(K), upper semicontinuous on K, and satisfies (3.2) and (3.3), then φ is the profile of a homogeneous distance that satisfy BCP on H. We give below an example of such a function that also illustrates the fact that such profiles need not be continuous on K.
where φ 1 is given by Example 2.7. It follows from the definition of φ 1 that φ 2 is continuous on D \ {(−1, 0), (1, 0)} but not at (−1, 0) and (1, 0). We prove now that φ 2 is the profile of a homogeneous distance that satisfies BCP. First, φ 2 is the sum of φ 1 with a non negative, continuous, and concave function on D. It follows from Example 2.7 that φ 2 is concave on int(D) and upper semicontinuous on D, hence φ 2 is the profile of a homogeneous distance by Proposition 2.5. Next, φ 2 ∈ C 2 (int(D)) with ∇φ(0) = 0, hence (3.3) holds. By Theorem 3.1, to prove that BCP holds for the homogeneous distance whose profile is given by φ 2 , it is thus sufficient prove that (3.2) holds.
Homogeneous distances that satisfy the assumptions of Theorem 3.1 can also be constructed as follows. Let K ⊂ R 2 be a compact convex set such that 0 ∈ int(K) and K = −K. Let ψ ∈ Lip(K) and assume that ψ satisfies (3.2) and (3.3). By [4, Proposition 6.3], there is a constant C ≥ 0 such that, for every c ≥ C, the function ψ + c : K → [0, +∞) defines the profile of a homogeneous distance on H. Since ψ + c satisfies (3.2) and (3.3) as well, it follows from Theorem 3.1 that such homogeneous distances satisfy BCP on H. This allows in particular to contruct another large class of homogeneous distances on H for which BCP holds.
Next, for rotationally invariant homogeneous distances, Theorem 3.1 can be rephrased as follows. We refer to the paragraph before Theorem 1.6 for the definition of rotationally invariant homogeneous distances and of their radial profile. Proof. Let φ : π(B) → [0, +∞) and ϕ : [−r d , r d ] → [0, +∞) denote respectively the profile and radial profile of d, so that φ(v) = ϕ( v ). We have together with the assumption that |ϕ (s)/s| is essentially bounded in a neighbourhood of 0.
Hence BCP holds in (H, d) by Theorem 3.1.
Note that, since the radial profile ϕ is even, then, if ϕ ∈ C 1 ((−r d , r d )) and ϕ is differentiable at 0, we have ϕ (0) = 0 and |ϕ (s)/s| is bounded in a neighbourhood of 0. This leads to the following corollary.
Corollary 3.9 gives the "if" part in Theorem 1.6. The validity of BCP for the rotationally invariant homogeneous distances d (1.8) when > 0 comes as an application of Corollary 3.9. Indeed, let > 0 be fixed and let B denote the unit ball centered at the origin in (H, d ). One can easily check that π(B ) is a Euclidean disc with radius r := 1/ √ 1 + 2 and that the radial profile ϕ of d is given by and hence, ϕ (s) ≤ ϕ (0) = − 2 /2 for every s ∈ (0, r ) . Therefore one can apply Corollary 3.9 with m = 2 /2 and BCP holds in (H, d ) for every > 0 as claimed.
Remark 3.10. The proof of Theorem 3.1 and Corollary 3.9 can also be used to prove the validity of BCP for the rotationally invariant continuous homogeneous quasi-distance on H defined by whose unit ball centered at the origin can be described as Remark 3.11. Corollary 3.9 can be used to recover the fact that the rotationally invariant homogeneous distances d α considered in [6] satisfy BCP.

3.2.
An abstract sufficient condition. We recall that a subset A of metric Theorem 3.12. Let d be a homogeneous distance on H. Assume that there are ∈ (0, 1), > 0, an -compact metric space (X, ρ) and a map σ : be a finite Besicovitch family of balls in (H, d), see Definition 1.1, and assume that r 1 ≥ · · · ≥ r n > 0. By left-invariance, we may also assume that 0 ∈ ∩ n i=1 B(p i , r i ), and, with no loss of generality, that r i = d(0, p i ) for every 1 ≤ i ≤ n.
We set i 1 := 1 and then iteratively i k+1 := min{j > i k : r j < r i k } as long as possible. We get from this construction a finite sequence i 1 = 1 < · · · < i l ≤ n for some integer l ≥ 1 with the following properties. Set i l+1 := n + 1 and I k := {i k , . . . , i k+1 − 1} for k ∈ {1, . . . , l}. First, for all k ∈ {1, . . . , l} and all j ∈ I k , r i k ≤ r j ≤ r i k . Second, r i k+1 < r i k for all k ∈ {1, . . . , l − 1} whenever l ≥ 2. Third, ∪ l k=1 I k = {1, . . . , n}. Next, we prove that, for all k ∈ {1, . . . , l}, Card I k ≤ C where C is the maximal cardinality of an -net in B. Recall that B is a compact subset of (H, d) and hence is -compact. For all j ∈ I k , d(0, p j ) = r j ≤ r i k and, for all i, j ∈ I k , i = j, d(p i , p j ) > max(r i , r j ) ≥ r i k . Hence {δ 1/r i k (p j )} j∈I k is an -net in B and the claim follows.
All together, we get Card B = j k=1 Card I k ≤ CC , and hence BCP holds in (H, d).
Remark 3.14. The proof of Theorem 3.12 only uses properties that the Heisenberg group shares with any other homogeneous group equipped with a homogeneous distance. Hence the abstract sufficient condition for the validity of BCP given in Theorem 3.12 holds more generally for homogeneous distances on homogeneous groups.
The proof of Theorem 3.1 will be based on the following corollary of Theorem 3.12.
3.3. Proof of Theorem 3.1. From now on in this section we assume that d is a homogeneous distance on H that satisfies the assumptions of Theorem 3.1. The validity of BCP will follow as an application of Corollary 3.15. Namely, we will prove the following lemma.
Before giving the proof of Lemma 3.16, we first explain how Theorem 3.1 follows from Lemma 3.16 in conjonction with Corollary 3.15.
Proof of Theorem 3.1. Let ∈ (0, 1), θ ∈ (0, π/2) be given by Lemma 3.16, and set := min( , If z 0 ≥ 0, the fact that δ t (q) ∈ B(p 0 , 1) for every t ∈ [0, ] follows from the first conclusion in Lemma 3.16. If z 0 < 0, then Therefore it follows from the second conclusion in Lemma 3.16 that δ t (q −1 ) · p 0 ∈ B, and hence The rest of the section is devoted to the proof of Lemma 3.16. It will be achieved through a series of lemmas, from Lemma 3.19 to Lemma 3.47 below.
We say that a constant is an absolute constant if its value can be chosen depending only on the parameters m, α, κ and M given by (3.2) and (3.3), geometric properties of the set K and properties of the function φ. We refer to Proposition 2.2 for these latter properties. In particular K ⊂ R 2 is bounded and we let D > 0 be such that We also set where κ is given by (3.3).
First, if ≤ α/2 is chosen small enough, we have .
We have where D is given by (3.17), which gives a contradiction.
Next, it follows from elementary computations from planar geometry that the assumption (v, w) ≤ together with (3.21) imply (v, v − w) ≤ α/2 and w − v ≤ v provided > 0 is chosen small enough, see Figure 1.
which concludes the proof.
By continuity of φ at s k v ∈ int(K), one can find ω k ∈ (0, 1/2k) such that On the other hand, one can find τ k > 0 such that, for every τ ∈ (0, τ k ), Then, choosing v such that v = s k v and ∠(v, v ) = τ , and setting z := φ(v ), we get that q : then v ∈ ∂K. By (2.4) one can find a positive increasing sequence (s k ) converging to 1 such that, for every k ≥ 1, where D is given by (3.17), and By continuity of φ at s k v ∈ int(K), one can find ω k ∈ (0, 1/3k) such that and one can find τ k > 0 such that, for every τ ∈ (0, τ k ), (3.25) holds. Then, for τ ∈ (0, τ k ), we let u τ ∈ S 1 be such that ∠(v, u τ ) = τ and we set v τ := r(u τ )u τ ∈ ∂K.
For x, y ∈ R, we use the notation x y to mean that there is an absolute constant C > 0 such that x ≤ Cy.
We need now to prove that (3.36) and (3.37) hold for every t ∈ [0, ] provided is chosen small enough.
Proof. First, note that since k(t) is a convex function, it follows from (3.39) that, for every t ∈ [0, 1], We now divide the proof into two cases.
If θ = 0 , then k(t) = |rt − a 0 | and we have for t = a 0 /r, if is chosen small enough, and we conclude the proof as in the case where θ = 0.

Necessary conditions
This section is devoted to the proof of necessary conditions for the validity of BCP on H equipped with a homogeneous distance. In Section 4.1, we recall a necessary condition proved in [6] and give a consequence of this condition on which our arguments will be mainly based, see Proposition 4.3. In Section 4.2, we prove Theorem 1.5, see Theorem 4.10, Theorem 4.16, and Theorem 4.17 for more detailed statements. In Section 4.3, we prove several improvements of the necessary conditions given in Section 4.2 under additional assumptions about the regularity of the profile of the homogeneous distance. Theorem 1.10 will come as a corollary of these improvements.

4.1.
Preliminaries. For technical convenience, we identify when needed in this section H with R 3 in the obvious way. Namely, we identify p = (v, z) ∈ H, where v = (x, y) ∈ R 2 and z ∈ R, with (x, y, z) ∈ R 3 .
For q ∈ H, u ∈ R 3 \ {0} and α ∈ (0, π/2), we denote by C(q, u, α) the open Euclidean half-cone with vertex q, axis q + R >0 u and opening 2α, namely, For q ∈ H, we denote by τ q the left-translation τ q (p) := q · p. When Given a homogeneous distance d on H with unit ball centered at the origin B, we say that u ∈ R 3 \ {0} points out of B at q ∈ ∂B if there exist an open neighbourhood U of q and α ∈ (0, π/2) such that For v ∈ R 2 \ {0} and > 0, we denote by Ω (v) the set of points q ∈ ∂B such that π(q) = 0, (v, π(q)) < and (τ q ) * (−v) points out of B at q.
The following sufficient condition for the non-validity of BCP can be found in [6]. Assume that one can find v ∈ R 2 \ {0} such that Ω (v) = ∅ for every > 0. Then BCP does not hold in (H, d).
Hence Ω (v) = ∅ implies Ω (v) = ∅ for every 0 < ≤ . Therefore, Theorem 4.1 can be rephrased as a necessary condition for the validity of BCP in the following way.
The major part of our arguments in Sections 4.2 and 4.3 will be based on the next proposition which is a consequence of Corollary 4.2.
Proposition 4.3. Let d be a homogeneous distance on H and assume that BCP holds in (H, d). Then, for all v ∈ R 2 \{0}, there exists v ∈ (0, π/4) such that the following holds. Let q ∈ ∂B be such that π(q) = 0 and (v, π(q)) < v . Assume that there is an open neighbourhood U of q and a continuous map F : U → R which is differentiable at q and such that F (q) = 0 and Proof. Let v ∈ R 2 \ {0} and v ∈ (0, π/4) be given by Corollary 4.2 so that Ω v (v) = ∅. Let q ∈ ∂B be as in the statement of the proposition, and in particular, such that π(q) = 0 and (v, π(q)) < v . Since Ω v (v) = ∅, (τ q ) * (−v) does not point out of B at q. Hence, for every k ∈ N * , one can find Since F is differentiable at q, it follows that and hence ∇F (q), (τ q ) * (v) ≥ 0.

4.2.
Necessary conditions. In this section, we let d denote a homogeneous distance on H with unit ball centered at the origin B and profile φ : K → [0, +∞) (see Proposition 2.2). We also set for w ∈ K and z ∈ R. In other words, we have We begin with a simple consequence of Proposition 4.3 for later use.
The first part of Theorem 1.5 can deduced from the next proposition, see also Theorem 4.10.
Proposition 4.8. Assume that BCP holds in (H, d). Then there is a countable closed set W ⊂ S 1 such that, for all u ∈ S 1 \ W , there are m u > 0, α u ∈ (0, π/2), and an open subset O u ⊂ S 1 containing u such that (4.9) ∇φ(w), w ≤ −m u w w for all w ∈ int(K) \ {0} such that φ is differentiable at w and w/ w ∈ O u , and for all w ∈ R 2 \ {0} such that (w, w ) < α u .
Proof. For v ∈ S 1 , let v ∈ (0, π/4) be given by Lemma 4.5 and set Then W := W + ∪ W − is clearly closed. To prove that W is countable, we first claim that Γ + w ∩ Γ + w = ∅ whenever w and w are distinct points in W + . Indeed assume that w and w are distinct points in S 1 such that Γ + w ∩ Γ + w = ∅. Then w = −w and we can assume with no loss of generality that ∠(w, w ) > 0. Let u ∈ Γ + w ∩ Γ + w . We have ∠(w , u) > 0. This implies that 0 < ∠(w, w ) < ∠(w, u) < w and hence w ∈ Γ + w . It follows that w ∈ W + which proves the claim. Thus {Γ + w } w∈W + is a family of disjoint open subsets of S 1 and hence is countable. One shows in a similar way that W − is countable.
Note that if w ∈ O u and w ∈ S 1 are such that (w, w ) < α u , then (u, w ) < δ u , and hence, w ∈ Γ + v ∩ Γ − v . Let w ∈ int(K)\{0} be such that φ is differentiable at w and w/ w ∈ O u , and let w ∈ S 1 be such that (w, w ) < α u . By the previous remark, we have w = a v + b v for some a > 0 and b > 0. Then Using (4.6) and (4.7), we obtain and Hence, since a > 0 and b > 0, we get ∠(a v − b v , w) ∈ (α u , π − α u ) , see Figure 2. Moreover, We gather in Theorem 4.10 below the necessary conditions that can be deduced from Lemma 4.5 and Proposition 4.8.  Moreover, there is a countable closed set W ⊂ S 1 such that S 1 \ W can be written as a countable union of relatively open subsets O j and, for each j, there are m j > 0 and α j ∈ (0, π/2) such that (4.12) ∇φ(w), w ≤ −m j w w for all w ∈ int(K) \ {0} such that φ is differentiable at w and w/ w ∈ O j , and for all w ∈ R 2 \ {0} such that (w, w ) ≤ α j .
Proof. The first claim (4.11) is a straightforward consequence of Lemma 4.5 when applied with v := w/ w for w ∈ int(K) \ {0} such that φ is differentiable at w. The next two claims (4.12) and (4.13) can be deduced from Proposition 4.8 by standard arguments.
For rotationally invariant homogeneous distances, Theorem 4.10 has the following consequence, Corollary 4.14 below. We refer to the paragraph before Theorem 1.6 for the definition of rotationally invariant distances and of their radial profile. Proof. Let φ denote the profile of d so that Since φ ∈ Lip loc (int(K)) and ϕ ∈ Lip loc ((−r d , r d )), for a.e. v ∈ S 1 , we have ∇φ(sv), v = ϕ (s) for a.e. s ∈ (0, r d ).
By (4.12), one can find such a v ∈ S 1 such that there is m > 0 such that ∇φ(sv), v ≤ −ms for a.e. s ∈ (0, r d ) which gives the conclusion.
The "only if" part in Theorem 1.6 is a consequence of Corollary 4.14, which can for instance be used to recover the fact that the Korányi distance (1.7) does not satisfy BCP. Indeed, the radial profile of the Korányi  where v := (x, y).
We go back now to general homogeneous distances d with profile φ : K → [0, +∞). As a simple consequence of Theorem 4.10 together with the fact that φ is locally Lipschitz on int(K), we get that φ achieves its maximum on K at the origin.
We first prove that (4.11) can be improved into a strict inequality at points w ∈ int(K) \ {0} where ∇φ is differentiable.
Proposition 4.22. Assume that BCP holds on (H, d). Let w ∈ int(K) \ {0} be such that φ is differentiable on a neighbourhood of w and ∇φ is differentiable at w. Then ∇φ(w), w < 0.
It follows that ω(v, w) = 0 for all v ∈ R 2 and we get a contradiction since w = 0.
We now turn to the study of the properties of the Hessian of the profile at the origin. First, we prove that if φ is assumed to be differentiable on a neighbourhood of 0 with ∇φ differentiable at 0, then Hφ(0) is semi-definite negative.
Proposition 4.25. Assume that BCP holds on (H, d). Assume that φ is differentiable on a neighborhood of 0 with ∇φ differentiable at 0. Then the Hessian Hφ(0) of φ at 0 is semi-definite negative.
Finally, we improve the previous proposition and show that Hφ(0) is definite negative if Hφ is assumed to be differentiable at 0. Proof. We know from Proposition 4.25 that Hφ(0) is semi-definite negative. By contradiction assume that one can find v ∈ R 2 \{0} such that Hφ(0)v = 0. Arguing as in the proof of Proposition 4.25, let w ∈ R 2 and let γ : I → int(K) be a smooth curve defined on some open interval I ⊂ R containing 0 such that γ(0) = 0 and γ (0) = v and γ (0) = w. We have g γ,v (0) = 0, g γ,v (0) = 0 and g γ,v (t) ≥ 0 for all t > 0 small enough with g γ,v differentiable at 0. This implies that g γ,v (0) ≥ 0 with Setting C := 2 ∇ Hφ (v), v (0), v , it follows that ω(w, v) ≥ C for all w ∈ R 2 , and we get a contradiction since v = 0.