Largest nearest-neighbour link and connectivity threshold in a polytopal random sample

Let $X_1,X_2, \ldots $ be independent identically distributed random points in a convex polytopal domain $A \subset \mathbb{R}^d$. Define the largest nearest neighbour link $L_n$ to be the smallest $r$ such that every point of $\mathcal X_n:=\{X_1,\ldots,X_n\}$ has another such point within distance $r$. We obtain a strong law of large numbers for $L_n$ in the large-$n$ limit. A related threshold, the connectivity threshold $M_n$, is the smallest $r$ such that the random geometric graph $G(\mathcal X_n, r)$ is connected. We show that as $n \to \infty$, almost surely $nL_n^d/\log n$ tends to a limit that depends on the geometry of $A$, and $nM_n^d/\log n$ tends to the same limit.


Introduction
This paper is primarily concerned with the connectivity threshold and largest nearestneighbour link for a random sample X n of n points specified compact region A in a ddimensional Euclidean space.
The connectivity threshold, here denoted M n , is defined to be the smallest r such that the random geometric graph G(X n , r) is connected.For any finite X ⊂ R d the graph G(X , r) is defined to have vertex set X with edges between those pairs of vertices x, y such that x − y ≤ r, where • is the Euclidean norm.More generally, for k ∈ N, the k-connectivity threshold M n,k is the smallest r such that G(X n , r) is k-connected (see the definition in Section 2).
The largest nearest neighbour link, here denoted L n , is defined to be the the smallest r such that every vertex in G(X n , r) has degree at least 1.More generally, for k ∈ N with k < n, the largest k-nearest neighbour link L n,k is the smallest r such that every vertex in G(X n , r) has degree at least k.These thresholds are random variables, because the locations of the centres are random.We investigate their probabilistic behaviour as n becomes large.
We shall derive strong laws of large numbers showing that that nL d n,k / log n converges almost surely (as n → ∞) to a finite positive limit, and establishing the value of the limit.Moreover we show that nM d n,k / log n converges to the same limit.These strong laws carry over to more general cases where k may vary with n, and the distribution of points may be non-uniform.We give results of this type for A a convex polytope.
Previous results of this type (both for L n,k and for M n,k ) were obtained for A having a smooth boundary, and for A a d-dimensional hypercube; see [5].It is perhaps not obvious from the earlier results, however, how the limiting constant depends on the geometry of ∂ A, the topological boundary of A, for general polytopal A, which is quite subtle.
It turns out, for example, that when d = 3 and the points are uniformly distributed over a polyhedron, the limiting behaviour of L n is determined by the angle of the sharpest edge if this angle is less than π/2.We believe (but do not formally prove here) that if this angle exceeds π/2 then the point of X n furthest from the rest of X n is asymptotically uniformly distributed over ∂ A, but if this angle is less than π/2 the location of this point in is asymptotically uniformly distributed over the union of those edges which are sharpest.
Our motivation for this study is twofold.First, understanding the connectivity threshold in dimension two is vital in telecommunications, for example, in 5G wireless network design, with the nodes of X n representing mobile transceivers (see for example [1]).Second, detecting connectivity is a fundamental step for detecting all other higher dimensional topological features in modern topological data analysis (TDA), where the dimension of the ambient space may be very high.See [2,3] for discussion of issues related to the one considered here, in relation to TDA.General motivation for considering random geometric graphs is discussed in [5].
While our main results are presented (in Section 2) in the concrete setting of a polytopal sample in R d , our proofs proceed via general lower and upper bounds (Propositions 3.2 and 3.6) that are presented in the more general setting of a random sample of points in a metric space satisfying certain regularity conditions.This could be useful in possible future work dealing with similar problems for random samples in, for example, a Riemannian manifold with boundary, a setting of importance in TDA.

Statement of results
Throughout this paper, we work within the following mathematical framework.Let d ∈ N. Suppose we have the following ingredients: • A finite compact convex polytope A ⊂ R d (i.e., one with finitely many faces).
• A Borel probability measure µ on A with probability density function f .
• On a common probability space (S, F , P), a sequence X 1 , X 2 , . . . of independent identically distributed random d-vectors with common probability distribution µ, and also a unit rate Poisson counting process (Z t ,t ≥ 0), independent of (X 1 , X 2 , . ..) (so Z t is Poisson distributed with mean t for each t > 0).
For n ∈ N, t > 0, let X n := {X 1 , . .., X n }, and let P t := {X 1 , . . ., X Z t }.These are the point processes that concern us here.Observe that P t is a Poisson point process in R d with intensity measure tµ (see e.g.[4]). For For any point set X ⊂ R d and any D ⊂ R d we write X (D) for the number of points of X in D, and we use below the convention inf(∅) := +∞.
Given n, k ∈ N, and t ∈ (0, ∞), define the largest k-nearest neighbour link L n,k by Set L n := L n,1 .Then L n is the largest nearest-neighbour link.
We are chiefly interested in the asymptotic behaviour of L n for large n.More generally, we consider L n,k where k may vary with n.
Let θ d := π d/2 /Γ(1 + d/2), the volume of the unit ball in R d .Given x, y ∈ R d , we denote by [x, y] the line segment from x to y, that is, the convex hull of the set {x, y}.
Given m ∈ N and functions f : Throughout this section, assume we are given a constant β ∈ [0, ∞] and a sequence k : We make use of the following notation throughout: 3) with Ĥ0 (0) := 0. Note that Ĥa (x) is increasing in x, and that Ĥ0 (x) = x and Ĥa (0) = a.Throughout this paper, the phrase 'almost surely' or 'a.s.' means 'except on a set of P-measure zero'.For n ∈ N, we use [n] to denote {1, 2, . .., n}.We write f | A for the restriction of f to A.
Let Φ(A) denote the set of all faces of the polytope A (of all dimensions up to d − 1).Also, let Φ * (A) := Φ(A) ∪ {A}; it is sometimes useful for us to think of A itself as a face, of dimension d.
Given a face ϕ ∈ Φ * (A), denote the dimension of this face by D(ϕ).Then 0 ≤ D(ϕ) ≤ d, and ϕ is a D(ϕ)-dimensional polytope embedded in R d .Let ϕ o denote the relative interior of ϕ, and set ∂ ϕ Then there is a cone K ϕ in R d such that every x ∈ ϕ o has a neighbourhood U x such that A ∩U x = (x + K ϕ ) ∩U x .Define the angular volume ρ ϕ of ϕ to be the d-dimensional Lebesgue measure of K ϕ ∩ B(o, 1).
For example, if If D(ϕ) = 0 then ϕ = {v} for some vertex v ∈ ∂ A, and ρ ϕ equals the volume of B(v, r) ∩ A, divided by r d , for all sufficiently small r.If d = 2, D(ϕ) = 0 and ω ϕ denotes the angle subtended by A at the vertex ϕ, then ρ ϕ = ω ϕ /2.If d = 3 and D(ϕ) = 1, and α ϕ denotes the angle subtended by A at the edge ϕ (which is the angle between the two boundary planes of A meeting at ϕ), then ρ ϕ = 2α ϕ /3.Theorem 2.1.Suppose A is a compact convex finite polytope in R d .Assume that f | A is continuous at x for all x ∈ ∂ A, and that f 0 > 0. Assume k(•) satisfies (2.2).Then, almost surely, Ĥβ (D(ϕ)/d) , max v∈V β ρ v f (v)  .
In particular, if β = 0 the above limit comes to max 3 4π f 0 , 1 π f 1 , max e∈E Corollary 2.4 ( [5]).Suppose A = [0, 1] d , and f | A is continuous at x for all x ∈ ∂ A. For 1 ≤ j ≤ d let ∂ j denote the union of all (d − j)-dimensional faces of A, and let f j denote the infimum of f over ∂ j .Assume (2.2) with β < ∞.Then It is perhaps worth spelling out what the preceding results mean in the special case where β = 0 (for example, if k(n) is a constant) and also µ is the uniform distribution on A (i.e.f (x) ≡ f 0 on A).In this case, the right hand side of (2.6) comes to max ϕ∈Φ * (A)

D(ϕ)
(d f 0 ρ ϕ ) .The limit in (2.7) comes to 1/(π f 0 ), while the limit in Corollary 2.3 comes to f −1 0 max[1/π, max e (1/(2α e ))].So far we have only presented results for the largest k-nearest neighbor link.A closely related threshold is the k-connectivity threshold defined by where a graph G of order n is said to be k-connected (k < n) if G cannot be disconnected by the removal of at most k − 1 vertices.Set M n,1 = M n .Then M n is the connectivity threshold.
Notice that for all k, n with k < n we have  Remark 2.7.Theorems 2.1 and 2.5 extend earlier work found in [5] on the case where A is the unit cube, to more general polytopal regions.The case where A has a smooth boundary is also considered in [5] (in this case with also k(n) = const., the result was first given in [6] for L n,k and in [7] for M n,k ).
Remark 2.8.In [8], similar results are given for the k-coverage threshold R n,k , which is given by (2.12) Our results here, together with [8, Theorem 4.2], show that both

Proofs
In this section we prove the results stated in Section 2. Throughout this section we are assuming we are given a constant β ∈ [0, ∞] and a sequence (k(n)) n∈N satisfying (2.2).
Recall that µ denotes the distribution of X 1 , and this has a density f with support A, and that L n,k is defined at (2.1).Recall that Ĥβ (x) is defined to be the y ≥ β such that yH(β /y) = x, where H(•) was defined at (2.4).
For n ∈ N and p ∈ [0, 1] let Bin(n, p) denote a binomial random variable with parameters n, p. Recall that H(•) was defined at (2.4), and Z t is a Poisson(t) variable for t > 0. The proofs in this section rely heavily on the following lemma.

A general lower bound
In this subsection we present an asymptotic lower bound on L n,k(n) , not requiring any extra assumptions on A. In fact, A here can be any metric space endowed with a Borel probability measure µ which satisfies the following for some ε ′ > 0 and some d > 0: The definition of L n,k at (2.1) carries over in an obvious way to this general setting.Later, we shall derive the results stated in Section 2 by applying the results of this subsection to the different regions within A (namely interior, boundary, and lower-dimensional faces).
Given r > 0, a > 0, define the 'packing number' ν(r, a) be the largest number m such that there exists a collection of m disjoint closed balls of radius r centred on points of A, each with µ-measure at most a.
Then, given n sufficiently large, we have ν(r n , ar d n ) > 0 so we can find y n ∈ A such that µ(B(y n , r n )) ≤ ar d n , and hence nµ( is binomial with parameters n and µ(B(y n , r n )), by Lemma 3.1(a) we have that , which is summable in n.Thus by the Borel-Cantelli lemma, almost surely event , so for all n large enough, we can (and do) choose Then by a simple coupling, and Lemma 3.1(a), Let δ ′ ∈ (0, 1).By (3.1), for n large enough and all x ∈ A, ), so by the preceding estimates, . Both of these upper bounds are summable in n, so by the Borel-Cantelli lemma, almost surely for all large enough n we have the event Suppose the above event occurs and suppose m ∈ N with z(n) ≤ m ≤ z(n + 1).Note that and this yields the result for this case.
by a simple coupling, and Lemma 3.1(e), Hence, by our choice of ε, there is a constant c > 0 such that for all large enough n and all i ∈ [m n ] we have Since x n,i ∈ A, by (3.1), for n large enough and 1 ) is binomially distributed with probability parameter bounded away from zero.Also max 1≤i≤m n E [P λ (n) (B(x n,i , r n ))] tends to infinity as n → ∞.Therefore there exists η > 0 such that for all large enough n, defining the event we have for all large enough n that inf Hence, setting E n := ∪ m n i=1 E n,i , for all large enough n we have By assumption and therefore P[E c n ] is is summable in n.By Lemma 3.1(d), and Taylor expansion of H(x) about x = 1 (see the print version of [5,Lemma 1.4] for details; there may be a typo in the electronic version), for all n large enough P ).If E n occurs, and Z λ − (n) ≤ n, and Z λ (n) ≥ n, then for some i ≤ m n there is at least one point of X n in B(x n,i , δ ′ r n ) and at most k(n) points of X n in B(x n,i , r n ), and hence L n,k(n) > (1 − δ ′ )r n .Hence by the union bound which is summable in n by the preceding estimates.Therefore by the Borel-Cantelli lemma, so the result follows for this case too.

Proof of Theorem 2.1
In this subsection we assume, as in Theorem 2.1, that A is a compact convex finite polytope in R d .We also assume that the probability measure µ has density f with respect to Lebesgue measure on R d , and that f | A is continuous at x for all x ∈ ∂ A, and that f 0 > 0, recalling from (2.3) that f 0 := ess inf x∈A f (x).Also we let k(n) satisfy (2.2) for some There exists ε ′ > 0 depending only on f 0 and A, such that (3.1) holds.
Proof.Let B 0 be a (fixed) ball contained in A, and let b denote the radius of B 0 .For x ∈ A, let S x denote the convex hull of B 0 ∪ {x}.Then S x ⊂ A since A is convex.If x / ∈ B 0 , then for r < b the set B(x, r) ∩ S x is the intersection of B(x, r) with a cone having vertex x, and since A is bounded the angular volume of this cone is bounded away from zero, uniformly over x ∈ A \ B 0 .Therefore r −d Vol(B(x, r) ∩ A) is bounded away from zero uniformly over r ∈ (0, b) and x ∈ A \ B 0 (and hence over x ∈ A).Since we assume f 0 > 0, (3.1) follows.
There is a constant c > 0 such that for small enough r > 0 we can find at least cr −D(ϕ) points x i ∈ B(x 0 , δ ) ∩ ϕ that are all at a distance more than 2r from each other, and therefore ν(r, aρ ϕ r d ) = Ω(r −D(ϕ) ) as r ↓ 0. Thus by Proposition 3.2 we have lim inf almost surely, and (3.4) follows.
If we assumed f | A to be continuous on all of A, we would not need the next lemma because we could instead use Lemma 3.4 for ϕ = A as well as for lower-dimensional faces.However, in Theorem 2.1 we make the weaker assumption that f | A is continuous at x only for x ∈ ∂ A. In this situation, we also require the following lemma to deal with ϕ = A. Lemma 3.5.It is the case that Proof of Theorem 2.1.First suppose β < ∞.It is clear from (2.1) and (2.12) that L n,k ≤ R n,k+1 for all n, k.Also by (2.2) we have (k(n) + 1)/ log n → β as n → ∞.Therefore using [8, Theorem 4.2] for the second inequality below, we obtain almost surely that lim sup Alternatively, this upper bound could be derived using (2.9) and the asymptotic upper bound on M n that we shall derive in the next section for the proof of Theorem 2.5.By Lemmas 3.5 and 3.4, we have a.s. that lim inf Ĥβ (D(ϕ)/d) and combining this with (3.8) yields (2.6).Now suppose β = ∞.In this case, again using the inequality L n,k ≤ R n,k+1 and [8, Theorem 4.2], we obtain instead of (3.8) that a.s.

A general upper bound
In this subsection we present an asymptotic upper bound for M n,k(n) .As we did for the lower bound in Section 3.1, we shall give our result (Proposition 3.6 below) in a more general setting; we assume that A is a general metric space endowed with two Borel measures µ and µ * (possibly the same measure, possibly not).Assume that µ is a probability measure and that µ * is a doubling measure, meaning that there is a constant c * (called a doubling constant for µ * ) such that µ * (B(x, 2r)) ≤ c * µ * (B(x, r)) for all x ∈ A and r > 0.
We shall require further conditions on A: an ordering condition (O), a condition on balls (B), a topological condition (T) and a geometrical condition (G) as follows: (O) There is a total ordering of the elements of A.
(B) For all x ∈ A and r > 0, the ball B(x, r) is connected.
(G) There exists δ 1 > 0, and K 0 ∈ (1, ∞), such that for all r < δ 1 and any x ∈ A, the number of components of A\B(x, r) is at most two, and if there are two components, at least one of these components has diameter at most K 0 r.
Given D ⊂ A and r > 0, we write D r for {y ∈ A : dist(y, D) ≤ r}.Also, let κ(D, r) be the r-covering number of D, that is, the minimal m ∈ N such that D can be covered by m balls centred in D with radius r.
As before, given µ we assume X 1 , X 2 , . . . to be independent µ-distributed random elements of A with the k-connectivity threshold M n,k defined to be the minimal r such that G(X n , r) is k-connected, with X n := {X 1 , . . ., X n }.Proposition 3.6 (General upper bound).Suppose that (A, µ, µ * ) are as described above and A satisfies conditions (O), (B), (T), (G).Let ℓ ∈ N and let d > 0. For each j ∈ [ℓ] let a j > 0, b j ≥ 0. Suppose that for each K ∈ N, there exists r 0 (K) > 0 such that for all r ∈ (0, r 0 (K)), there is a partition {T ( j, K, r), j ∈ [ℓ]} of A with the following two properties.Firstly for each fixed K ∈ N, j ∈ [ℓ], we have and secondly, for all K ∈ N, j ∈ [ℓ], r ∈ (0, r 0 (K)) and any G ⊂ A intersecting T ( j, K, r) with diam(G) ≤ Kr, we have Assume (2.2).Then, almost surely, Later we shall use Proposition 3.6 in the case where A is a convex polytope in R d to prove Theorem 2.5, taking µ to be the measure with density f and taking µ * to be the restriction of Lebesgue measure to A (in fact, if f is bounded above then we could take µ * = µ instead).The sets in the partition each represent a region near to a particular face ϕ ∈ Φ * (A) (if ϕ = A the corresponding set in the partition is an interior region).In this case, coefficients a j in the measure lower bound (3.12) depend heavily on the geometry of the determining cone near a particular face.
As a first step towards proving Proposition 3.6, we spell out some useful consequences of the measure doubling property.In this result (and again later) we use | • | to denote the cardinality (number of elements) of a set.Lemma 3.7.Let µ * be a doubling measure on the metric space A, with doubling constant c * .We have the following.
(ii) For all r ∈ (0, 1) and all D ⊂ A, we can find L ⊂ D with |L | ≤ κ(D, r/5), such that D ⊂ ∪ x∈L B(x, r), and moreover the balls B(x, r/5), x ∈ L , are disjoint.Proof.To prove (i), let x ∈ A, r > 0. By the Vitali covering lemma, we can find a set U ⊂ B(x, r) such that balls B(y, εr/5), y ∈ U are disjoint and that B(x, r) ⊂ ∪ y∈U B(y, εr).Given countable σ ⊂ A, r > 0 and k ∈ N, we say that σ is (r, k)-connected if the geometric graph G(σ , r) is k-connected.Assuming condition (B) holds, we see that σ is (r, 1) connected if and only if σ r/2 is a connected subset of A. Then the number of (ar, 1)-connected subsets of L containing x 0 with cardinality n is at most c n , where c depends only on ℓ, a and c * .
Choose n 0 ∈ N such that r n < r 0 (k) for all n ∈ N with n ≥ n 0 .By Lemma 3.7-(i), for each j ∈ [ℓ] and n ∈ N we can find a set For and diam(σ ) ≤ 2Kr n .The claim about cardinality follows from this.Now we show (3.14).By Lemma 3.7-(i), for n large and for all x ∈ A, we can cover B(x, 2Kr n ) by ρ(ε/(10K)) balls of radius r n ε/5, and each of these balls contains at most ℓ points of L n .
For n ≥ n 0 and σ ⊂ L n , set and the claim follows.Also, provided n is large enough, we have k(n) ≤ k ′ (z(m(n))).Thus we have the event inclusions By (3.15), for any n ∈ N and σ ⊂ L n we have Hence by (3.12), for all large enough n and all σ ∈ ∪ j∈ A simple coupling shows that, provided m is large, we have By Lemma 3.1(b) and our choice of r n and ε, provided m is large, we have which is summable in m.
It follows from the Borel-Cantelli lemma that almost surely occurs only for finitely many m which implies that E n (K, u) occurs for only finitely many n.This completes the proof of the case β < ∞.Now assume β = ∞.For the rest of the proof assume also that ε ∈ (0, 1) is such that ua j (1 − ε) d > 1 for all j ∈ [ℓ].We do not have to go through the subsequence argument as before because the growth of k(n) is super-logarithmic.Now redefine where the second sum is over all possible shapes σ ⊂ L n of cardinality q that are (2εr n , 1)connected.Since every point in A is covered at most ℓ times, by (3.12) (with G = {z}), there exists ε 1 ∈ (0, 1) such that µ(σ εr n /5 ) ≥ (1/ℓ) ∑ z∈σ µ(B(z, εr n /5)) ≥ (q/ℓ)ε 1 (εr n /5) d .
As before, we can choose K = K 1 (large) so that the H(•) term in every summand is bounded from below by 1/2.By the super-logarithmic growth of k(n), we conclude that P[H n (K, u)] ≤ n −2 provided n is large, so that the Borel-Cantelli lemma gives the result in this case too.
Proof of Proposition 3.6.
and set r n := (u(logn)/n) 1/d .By Lemmas 3.10 and 3.11, there exists K ∈ N such that almost surely, E n (K, u) ∪ H n (K, u) occurs for at most finitely many n.By Lemma 3.9, if M n,k > r n then E n (K, u) ∪ H n (K, u) occurs.Therefore M n,k(n) ≤ r n for all large enough n, almost surely, and the result follows.

Proof of Theorem 2.5
In this subsection we go back to the mathematical framework in Section 2; that is, we make the assumptions in the statement of Theorem 2.5.In particular we return to assuming A is a convex polytope in R d with d ≥ 2, and the probability measure µ has a density f .We shall check the conditions required in order to apply Proposition 3.6.
To check these conditions, we shall use the following lemma and notation.
Points of A can be ordered by using the lexicographic ordering inherited from R d , thus (O).Since A is convex, for all x ∈ A and r > 0 the set B(x, r) ∩ A is convex and hence connected, implying (B).All convex polytopes are simply connected, and therefore unicoherent [5, Lemma 9.1], hence (T).Condition (G) follows immediately from Proposition 3.14, which we prove below.Proposition 3.14.Let A be a convex finite polytope in R d .Let N(•) denote the number of components of a set.There exists δ 1 > 0 such that for any x ∈ A any r ∈ (0, δ 1 ), we have N(A \ B(x, r)) ≤ 2.Moreover, in the case that N(A \ B(x, r)) = 2, the diameter of the smaller component is at most cr, where c is a constant depending only on A.
To apply Proposition 3.6, we need to define a partition of A for each small r > 0, then estimate the corresponding covering numbers and µ-measures in (3.12).
Taking into account a variety of boundary effects near ∂ A, one should consider separately regions near different faces of A. It is however not trivial to construct this partition in such a way that we can obtain tight µ-measure estimates in (3.12).The matter is complicated by the fact that the set G in (3.12) that intersects a region near ϕ is potentially close to a lower dimensional face lying inside ∂ ϕ.We can avoid the boundary complications by constructing inductively from regions near to the highest dimensional face to the lowest, with increasing 'thickness'.The partition made of T (ϕ, r)'s defined below and the left-over interior region is defined for this purpose.
Let (K j ) j∈N be an increasing sequence with K 1 = 1, and with K j+1 > (2K(A) + 1)K j for each j ∈ N.For instance, we could take K j = (2K(A) + 2) j−1 .Now for each r > 0 and ϕ ∈ Φ(A), define the set where the T stands for 'territory'.Also define T (A, r) := A \ ∪ ϕ∈Φ(A) ϕ rK d−D(ϕ) For each ϕ ∈ Φ * (A), we have T (ϕ, r) = ∅ for all r sufficiently small.Hence, there exists r 0 > 0 such that for all ϕ and all r < r 0 , T (ϕ, r) = ∅.Moreover, territories of distinct faces are disjoint, as we show in the following lemma.
As a last ingredient for applying Proposition 3.6, for each J > 1 and r ∈ (0, 1), we construct a partition of A and show (3.12) for all G with diameter at most Jr.The coefficients a j depend on the location of G in relation to faces of A. Proof.Item (i) follows by the definition of T (ϕ, r).Indeed, ϕ is contained in a bounded region within a D(ϕ)-dimensional affine space, and therefore can be covered by O(r −D(ϕ) ) balls of radius r.If we then take balls of radius r(1 + 2JK d−D(ϕ) ) with the same centres, they will cover T (ϕ, 2Jr), and one can then cover each of the larger balls a fixed number of balls of radius r.For (ii), let G ⊂ A with diam(G) ≤ Jr. Suppose first that G ∩ T (ϕ, 2Jr) = ∅ for some ϕ ∈ Φ(A).Let x 0 ∈ G ∩ T (ϕ, 2Jr).Then G r ⊂ B(x 0 , 2Jr).By Lemma 3.15, we see that B(x 0 , 2Jr) does not intersect any ϕ ′ ∈ Φ(A) with ϕ \ ϕ ′ = ∅.It follows that B(x 0 , 2Jr) ∩ A = B(x 0 , 2Jr) ∩ (z 0 + K ϕ ) (3.21) where K ϕ is the cone determined by ϕ and z 0 is the point of ϕ closest to x 0 .Set D(x, r) := B(x, r) ∩ (x + K ϕ ).We claim that for any x ∈ G, we have D(x, r) ⊂ A. Indeed, given y ∈ D(x, r), we can write y = z 0 + (x − z 0 ) + (y − x) =: z 0 + θ 1 + θ 2 .
It follows that (with ⊕ denoting Minkowski addition) In this case f ϕ = f 0 and ρ ϕ = θ d , and the claim (3.20) follows in this case too, completing the proof of (ii).
Proof of Theorem 2.5.By (2.9), and Theorem 2.1, it suffices to prove the upper bound.We shall do this by applying Proposition 3.6 in the situation of Theorem 2.5.
By Lemma 3.13, the restriction to A of Lebesgue measure has the doubling property, and conditions (O), (B), (T) and (G) are satisfied To apply Proposition 3.6, we need to define (for each K ∈ N and each r ∈ (0, r 0 (K))) a finite partition {T ( j, K, r)}.For this we take the sets T (ϕ, 2Kr), ϕ ∈ Φ * (A).By Lemma 3.15, and the definition of T (A, r), for each K ∈ N there exists r 0 (K) > 0 such that for r ∈ (0, r 0 (K)) the sets T (ϕ, 2Kr), ϕ ∈ Φ * (A), do indeed partition A.
For each ϕ ∈ Φ * (A), using Lemma 3.16-(i) we have the condition (3.11) in Proposition 3.6, where the constant denoted b j there is equal to D(ϕ).Also, using Lemma 3.16-(ii) we have the condition (3.12) in proposition 3.6, where the constant denoted a j there is equal to (1 − ε) f ϕ ρ ϕ .
.6)In the next three results, we spell out some special cases of Theorem 2.1.