Estimation of surface area

We study the problem of estimating the surface area of the boundary $\partial S$ of a sufficiently smooth set $S\subset\mathbb{R}^d$ when the available information is only a finite subset $\X\subset S$. We propose two estimators. The first makes use of the Devroye--Wise support estimator and is based on Crofton's formula, which, roughly speaking, states that the $(d-1)$-dimensional surface area of a smooth enough set is the mean number of intersections of randomly chosen lines. For that purpose, we propose an estimator of the number of intersections of such lines with support based on the Devroye--Wise support estimators. The second surface area estimator makes use of the $\alpha$-convex hull of $\X$, which is denoted by $C_{\alpha}(\X)$. More precisely, it is the $(d-1)$-dimensional surface area of $C_\alpha(\X)$, as denoted by $|C_\alpha(\X)|_{d-1}$, which is proven to converge to the $(d-1)$-dimensional surface area of $\partial S$. Moreover, $|C_\alpha(\X)|_{d-1}$ can be computed using Crofton's formula. Our results depend on the Hausdorff distance between $S$ and $\X$ for the Devroye--Wise estimator, and the Hausdorff distance between $\partial S$ and $\partial C_{\alpha}(\X)$ for the second estimator.


On surface area and length estimation
The estimation of surface areas has been extensively considered in stereology (see, for instance, [8,9] and [27]). It has also been studied as a further step in the theory of nonparametric set estimation (see [32]), and has practical applications in medical imaging (see [19]). In addition, the estimation of a surface area is widely used in magnetic resonance imagining techniques (see [28]).
The three-and two-dimensional cases are addressed in [10], which proposed parametric estimators when the available data are the distances to S from a sample outside the set but at a distance smaller than a given R > 0.
The two-dimensional case has many important applications. This is also true of the three-dimensional case. For instance, surface area is an important biological parameter in organs such as the lungs (see, for instance, [40]). The higher dimensional study is also important, at least from a theoretical point of view, because in [36] it is shown that the boundary surface plays an important role as a parameter of a probability distribution, which allows us to apply plug-in methods. To our knowledge, the only paper that tackles the surface area estimation problem in any dimension, when only "inside" data are available, is [22] and no convergence rates are given.
When, as in image analysis, one can observe n data points from two distinguishable sets of random data-points (one from inside S and the other from outside S), then the estimation of the surface area of the boundary has been tackled, for any d ≥ 2, in [19,21,29,32] and [41]. The proposals given in [19,32] and [21] aim to estimate the Minkowski content of ∂S. In [21], a very general convergence result is obtained, and in [19] a convergence rate of order n −1/2d is obtained under some mild hypotheses, and later on, in [32], a convergence rate of order (log(n)/n) 1/(d+1) is achieved under stronger assumptions. In [29], a very nice fully data-driven method that is based on the Delaunay triangulation is proposed under an homogeneous point process sampling scheme. The asymptotic rate of convergence of the variance is given but there is no global convergence rate because no result is obtained for the bias. Finally, in [41], a parameter-free procedure that is based on the Voronoi triangulation is proposed and a rate of convergence of order λ −1/d is obtained under a Poisson Point Process (PPP) sampling scheme (where λ is the intensity of the PPP).

Roadmap
When S ⊂ R d is a compact set, we aim to estimate its surface area; that is, the (d − 1)-Hausdorff measure of its boundary ∂S.
We propose two surface area estimators, at any finite dimension, when the available data is only a finite set X n ⊂ S. In this setting, the twodimensional case has mostly been studied. Assuming that X n is an iid sample, the convex case was first addressed in [11] (using Crofton's formula). Later on, under the α-convexity assumption, [5] obtained the convergence of the α-shape's perimeter to the perimeter of the support and the associated convergence rates are derived. When the data are given by a trajectory from a reflected Brownian motion (RBM) (with or without drift), a consistency result is obtained in Theorem 4 in [13].
Proposed estimator relies on Crofton's formula, which was proven in 1868 for convex subsets of R 2 and extended to arbitrary dimensions (see [39]). It states that the surface area of ∂S equals the integral of the number of intersections with ∂S of lines in R d (see Equations (3) and (4) for explicit versions of Crofton's formula for d = 2 and d ≥ 2, respectively).
The first proposed estimator is based on the Devroye-Wise support es- see [23], where n is the cardinality of X n , ε n → 0 as n → ∞ and B(X i , ε n ) denotes the closed ball in R d centred at X i and of radius ε n > 0. By use of S εn (X n ) andŜ 4εn (X n ) we propose an estimator of the number of intersection of a line with ∂S. The reader should be aware that this estimator is not just a just a plug-in method because (in general) the number of intersections of a line with ∂Ŝ εn (X n ) may not converge to the number of intersections of that line with ∂S. The main results regarding this estimator are stated in subsection 2.3 where it is proven that this estimator converges at a rate that is proportional to d H (X n , S) 1/2 (where d H denotes the Hausdorff distance). This rate can be improved to d H (X n , S) when adding a reasonable assumption on the shape of ∂S. These rates are known when X n is an iid sample, see Corollary 2. The computational aspects of this estimator are studied in subsection 2.4. The second uses the α-convex hull support estimator see [37], whereB(x, α) c denotes the complement of the open ball in R d centred at x and of radius α > 0. First we extend the results in [20]. More precisely, we prove that, in any dimension, the surface area of the hull's boundary-that is, |∂C α (X n )| d−1 -converges to |∂S| d−1 . This result is interesting in itself but in practice it is difficult to compute |∂C α (X n )| d−1 , especially for dimension d > 2. However, we will see that by means of Crofton's formula it can easily be estimated via the Monte Carlo method. The approach based on the α-convex hull is introduced in Section 3. A discussion of the rates of convergence is given in Section 4 and an algorithm based on the Monte Carlo method for the estimator based on the α-hull is introduced in Section 5. These results can be applied to many deterministic or random situations to obtain explicit convergence rates. We focus on two random situations: the case X n = {X 1 , . . . , X n } of iid drawn on S (with a density bounded from below by a positive constant), and the case of random trajectories of reflected diffusions on S. In particular, we provide convergence rates when the trajectory is the result of a RBM (see [13,14]). This last setting has several applications in ecology, where the trajectory is obtained by recording the location of an animal (or several animals) living in an area S, which is called its home range (the territorial range of the animal), and X t represent the position at time t transmitted by the instrument (see, for instance, [7,13,14], and the references therein).
The rate of convergence of the surface area estimator based onŜ εn (X n ), when X n is an iid sample, is of order n −1/2d . This can be improved to n −1/d , depending on the assumptions on the smoothness of ∂S. With the estimation of the support that uses the α-convex hull, when X n is an iid sample, we obtain a rate of order n −2/(d+1) .

Notations
Given a set S ⊂ R d , we denote byS, S and ∂S the interior, closure and boundary of S, respectively, with respect to the usual topology of R d . We We denote by B d (x, ε) (or sometimes just B(x, ε)) the closed ball in R d , of radius ε, centred at x, and ω d = |B d (x, 1)| d . Given two compact nonempty sets A, C ⊂ R d , the Hausdorff distance between A and C is defined by Given M a sufficiently smooth (d − 1)-manifold and x ∈ M , the affine tangent space of M at x is denoted by T x M . When S ⊂ R d is regular (i.e., compact and satisfying S =S) and has a C 1 regular boundary ∂S, then for any x ∈ ∂S we can define η x the outward normal unit vector at x; that is, the unit vector of (T x ∂S) ⊥ such that, for t > 0 small enough, Given a vector θ ∈ (S + ) d−1 and a point y, r θ,y denotes the line {y + λθ, λ ∈ R} = y + Rθ. If y 1 and y 2 are two points in r θ,y , then y i = y + λ i θ. With a slight abuse of notation, we write y 1 < y 2 when λ 1 < λ 2 .

Crofton's formula
In 1868, Crofton proved the following result (see [17]): given a convex set in the plane, whose boundary is denoted by γ, then its length |γ| 1 can be computed by n γ (θ, p) being the number of intersections of γ with the line r θ * ,θp , where θ * ∈ (S + ) 1 is orthogonal to θ, and dpdθ is the two-dimensional Lebesgue measure (see Figure 1). This result has been generalized to compact (not necessarily convex) sets in R d for any d > 2, and also to Lie groups (see [39]). To introduce the general Crofton formula in R d for a compact (d − 1)dimensional manifold M , let us define first the constant where Γ stands for the well-known Gamma function. Let θ ∈ (S + ) d−1 . Then, θ determines a (d − 1)-dimensional linear space θ ⊥ = {v : v, θ = 0}. Given y ∈ θ ⊥ , let us write n M (θ, y) = #(r θ,y ∩ M ), where # is the cardinality of the set (see Figure 2).
It is proven in [26] (see Theorem 3.2.26) that if M is a (d−1)-dimensional rectifiable set, then the integralgeometric measure of M (which will be denoted by I d−1 (M ), and is defined by the right-hand side of (4)) equals its (d − 1)-dimensional Hausdorff measure; that is, The measure dθ is the uniform measure on (S + ) d−1 (with total mass 1) and µ d−1 is the (d − 1)-dimensional Lebesgue measure.

Restrictions on the shape
We will now recall some well-known restrictions that are put on the shape in the set estimation.
When S is α-convex, a natural estimator of S from a random sample X n of points (drawn from a distribution with support S), is C α (X n ) (see [37]). Definition 1.2. A set S ⊂ R d is said to satisfy the outside α-rolling condition if for each boundary point s ∈ ∂S there exists an x ∈ S c such that B(x, α) ∩ ∂S = {s}. A compact set S is said to satisfy the inside α-rolling condition if S c satisfies the outside α-rolling condition.
Following the notation in [25], let Unp(S) be the set of points x ∈ R d with a unique projection on S. Definition 1.3. For x ∈ S, let reach(S, x) = sup{r > 0 :B(x, r) ⊂ Unp(S) . The reach of S is defined by reach(S) = inf reach(S, x) : x ∈ S , while S is of positive reach if reach(S) > 0. Remark 1. Throughout this paper, we assume that ∂S is the boundary of a compact set S ⊂ R d such that S =S. We also assume that S fulfills the outside and inside α-rolling conditions, and then ∂S is rectifiable (see Theorem 1 in [42]). From this it follows that I d−1 (∂S) = |∂S| d−1 < ∞, which implies (by (4)) that, except for a set of measure zero with respect to dµ d−1 (y)dθ, any line r θ,y meets ∂S a finite number of times: n ∂S (θ, y) < ∞. From Theorem 1 in [42], it also follows that ∂S is a C 1 manifold, which allows us for each x ∈ ∂S to define its unit outward normal vector η x .
For the estimator of the surface area based on the Devroye-Wise estimator, we will assume that ∂S satisfies a technical hypothesis, which is referred to as (C, ε 0 )-regularity.
• If ∂S is (C, ε 0 )-regular for some ε 0 > 0, then we will say that ∂S is C-regular.
Once the rolling balls condition is imposed, we will show through some examples in Figure 3 that the (C, ε 0 )-regularity of the boundary is quite mild.
(b) The second set, which is presented in Figure 4, is a two-dimensional 'peanut' that is made of 4 circular arcs. For all θ and ε small enough, we have ϕ θ (ε) = 2c θ ε where c θ is the number of connected components of F θ , which is less than 6, from which it follows that S has a 12-regular boundary.
(c) The third set, presented in Figure 5, is the surface of revolution generated by (b). Here we have that for all θ, E θ is a one-dimensional manifold with less than three connected components. The maximal length of a component is bounded by L, the length of the maximal perimeter (shown in blue in the figure). The reach of each E θ is (uniformly in θ) lower bounded by α > 0. All of these assertions allow us to claim that ∂S is 6L-regular. (c) three-dimensional peanut Figure 6: (d) an 'infinite wave' shape (d) The rolling ball condition is not sufficient to guarantee the (C, ε 0 )regularity of the boundary: this fails if, for instance, S is such that ∂S = S 1 ∪ S 2 ∪ S 3 ∪ S 4 (see Figure 6) with: It can easily be proven that such a set satisfies the rolling ball condition for any r 0 ≤ 1/80 but ϕ 0 (ε) → +∞ when ε → 0, which implies that ∂S is not C-regular.
For the Devroye-Wise type estimator, we will also show that the convergence rate can be quadratically improved if we additionally assume that the number of intersections between any line and ∂S is bounded from above (this excludes the case of a linear part in ∂S, such as in Figure 3). Definition 1.5. Given S ⊂ R d , we say that ∂S has a bounded number of linear intersections if there exists an N S such that for all θ ∈ (S + ) d−1 and y ∈ θ ⊥ , n ∂S (θ, y) ≤ N S .

Remark 2.
The previous definition can be replaced with a weaker requirement by asking that ∂S has a bounded number of linear intersections for almost all lines with respect to µ d−1 (y)dθ, and the corresponding results remain true.
2 Surface area estimation based on the Devroye-Wise estimator

A conjecture on the Devroye-Wise estimator
Since the set S is in general unknown, we first propose the natural plug-in idea of computing |∂Ŝ| d−1 , whereŜ is an estimator of S. There are several kinds of set estimators, depending on the geometric restrictions imposed on S and the structure of the data (see [13,23] and references therein). One of the most studied in the literature, which is also universally consistent, is the Devroye-Wise estimator (see [23]) that was introduced in (1). This all-purpose estimator has the advantage that it is quite easy to compute the intersection of a line with its boundary, as follows: Given a line r θ,y , we can compute Y i = ∂B(X i , ε n ) ∩ r θ,y , and then Z i = {y ∈ Y i , d(y, X n ) ≥ ε n }, so we have that, with probability one, Indeed, suppose, on the contrary, that there exists a z ∈ ∪ i Z i and z ∈S εn , then we have d(z, X n ) = ε n and z ∈ H{X i , d(X i , z) = ε n } (where H(E) is the convex hull of E). Thus, there are at least d + 1 observations on the same hypersphere of given radius ε n , but this event has probability 0 (see [31]). We conjecture that the plug-in estimator |∂Ŝ εn (X n )| d−1 satisfies the following: 1. If ε n < d H (X n , S), then ∂Ŝ εn (X n ) does not converge to ∂S and |∂Ŝ εn (X n )| d−1 does not converge to |∂S| d−1 .

A surface estimator based on the Devroye-Wise estimator
The aim of this section is to propose an estimator for the surface area based on the Devroye-Wise support estimator and Crofton's formula that can attain a convergence rate of order d H (X, S). The whole procedure is defined for any set X, but is not necessarily finite because we will apply our estimator to the case in which X is the trajectory of a Brownian motion. If X is not finite, then for a given ε > 0, we writeŜ ε (X) = B(X, ε). This procedure replaces n ∂S (θ, y) byn ε,X (θ, y) introduced in Definition 2.1, and then integratesn ,X (θ, y) as in Crofton's formula (see (5)). We will prove that (see Remark 4) by the (C, ε 0 )-regularity of the boundary, with probability one, r θ,y is not included in any (d−1)-dimensional affine tangent space (tangent to ∂S). Then, n ∂S (θ, y) = 2k S (θ, y), where k S (θ, y) is the number of connected components of r θ,y ∩ S.
Definition 2.1. Let ε be a positive real number and X ⊂ S be a set (not necessarily finite). Consider a line r θ,y . IfŜ ε (X) ∩ r θ,y = ∅, then definê n ε,X (θ, y) = 0. If not, then: • denote by I 1 , . . . , I m the connected components ofŜ ε (X) ∩ r θ,y . Order this sequence in such a way that • If for some consecutive intervals • Let j be the number of disjoint open intervals A 1 , . . . , A j that this process ended with. Then definen ε,X (θ, y) = 2j.
To roughly summarize this, we consider the connected components of S ε ∩ r θ,y and then 'link or glue' the ones that are in the same connected component ofŜ 4ε ∩ r θ,y . In the sequel, we will refer to this process as the gluing procedure.
To gain some insight into the relationship betweenn ε,X (θ, y) and n ∂Ŝε(X) (θ, y), observe thatn ε,X (θ, y) ≤ n ∂Ŝε(X) (θ, y). We also have thatn ε,X (θ, y) ≤ n ∂Ŝ 4ε (X) (θ, y). Indeed, let C 1 , . . . , C K be the connected components of r θ,y ∩ S 4ε and note that: 3. If d(C i , X) ≤ ε, then there exists an I j ⊂ C i and all the I j such that I j ⊂ C i are glued by the proposed procedure. Thus, there exists a unique j such that A j ⊂ C j .
Our first proposed estimator iŝ Under the assumption that ∂S has a bounded number N S of linear intersections (see Definition 1.5), we will consider, for a given N 0 ≥ N S ,

Main results on the Devroye-Wise based estimator.
Theorem 2.2. Let S ⊂ R d be a compact set fulfilling the outside and inside α-rolling conditions. Assume also that S is (C, ε 0 )-regular for some positive constants C and ε 0 . Let X n = {X 1 , . . . , X n } ⊂ S. Let ε n → 0 be such that d H (X n , S) ≤ ε n . Then, Moreover, for n large enough, The idea of the proof of Theorem 2.2 consists of proving that our algorithm allows a perfect estimation of n ∂S (θ, y) for the lines that are 'far enough' (fulfilling L(ε) for some ε > 0) from the tangent spaces. For the rest of the lines, we will prove in Corollary 5 that, under (C, 0 )-regularity, the integral ofn εn,Xn (θ, y) on the set of these lines, is bounded from above by C ε 1/2 n , C being a positive constant. Roughly speaking, a line fulfilling condition L(ε) does not meet the estimator ∂Ŝ n too many times.
From Theorem 2.2 and Theorem 4 in [18], we can obtain the rate of convergence for the iid case: Corollary 1. Let S ⊂ R d be a compact set fulfilling the inside and outside α-rolling conditions. Assume also that S is (C, ε 0 )-regular for some positive constants C and ε 0 . Let X n = {X 1 , . . . , X n } be the set of observations of an iid sample of X with distribution P X supported on S. Assume that P X has density f (w.r.t. µ d ) bounded from below by some c > 0. Let ε n = C (ln(n)/n) 1/d and C > (6/(cω d )) 1/d . Then, with probability one, for n large enough,Î As mentioned in Section 5.2 in [18], if ε n = 2 max i min j =i ||X i − X j ||, then with probability one, for n large enough, ε n ≤ 2d H (X n , S), which together with Corollary 1, entails that, with the aforementioned choice for ε n , our proposal is fully data driven, for the iid case.
If the number of linear intersections of ∂S is assumed to be bounded by a constant N S , the use of min(n εn , N 0 ) (for any N 0 ≥ N S ) allows us to obtain better convergence rates. Theorem 2.3. Let S ⊂ R d be a compact set fulfilling the outside and inside α-rolling conditions. Assume also that S is (C, ε 0 )-regular for some positive constants C and ε 0 , and that the number of linear intersections of ∂S is bounded by N S . Let X n = {X 1 , . . . , X n } ⊂ S. Let ε n → 0 be such that d H (X n , S) ≤ ε n and N 0 ≥ N S . Then, As before, we give the convergence rate associated to the iid setting and the RBM hypothesis as two corollaries of Theorem 2.3.
Corollary 2. Let S ⊂ R d be a compact set fulfilling the inside and outside α-rolling conditions. Assume also that S is (C, ε 0 )-regular for some positive constants C and ε 0 , and that ∂S has a bounded number of linear intersections. Let X n = {X 1 , . . . , X n } be the set of observations of an iid sample with distribution P X , supported on S. Assume that P X has density f (w.r.t. µ d ) bounded from below by some c > 0. Let ε n = C (ln(n)/n) 1/d and C > (6/(cω d )) 1/d . Then, with probability one, for n large enough, Here again, the choice of ε n = 2 max i min j ||X i − X j || is suitable but now the price to pay is the selection of the parameter N 0 .
In a more general setting, the conclusion of Theorem 2.3 holds when the set of points X n is replaced by the trajectory X T of any stochastic is well defined, even when X T is not a finite set (see Definition 2.1). We will assume that S is bounded with connected interior and ∂S is C 2 . This is the case (for example) of some reflected diffusions, and in particular the RBM. This has recently been proven in Corollary 1 in [13] for RBM without drift (see also [14] and [15] for the RBM with drift). The definition of an RBM with drift is as follows: given a d-dimensional Brownian motion {B t } t≥0 departing from B 0 = 0 and defined on a filtered probability space (Ω, F, {F t } t≥0 , P x ), an RBM with drift is the (unique) solution to the following stochastic differential equation on S: where the drift, ∇ f (x), is given by the gradient of a function f and is assumed to be Lipschitz, {ξ t } t≥0 is the corresponding local time; that is, a one-dimensional continuous non-decreasing process with ξ 0 = 0 that satisfies ξ t = t 0 I {Xs∈∂S} dξ s . Since the drift is given by the gradient of a function and S is compact, we have that its stationary distribution has a density bounded from below by a constant.
Corollary 3. Let S ⊂ R d be a non-empty compact set with connected interior such that S =S, and suppose that S fulfills the outside and inside α-rolling conditions. Assume also that S is (C, ε 0 )-regular for some positive constants C and ε 0 and that the number of linear intersections of ∂S is bounded by N S . Let X T ⊂ S be as before. Then, with probability one, for T large enough,Î

The algorithm
We will now describe an algorithm to computen ε,Xn (θ, y) for a given (θ, y), when the input is a finite set of n elements and ε > 0. For a reflected diffusion, we take X n ⊂ X T to be a dense enough subset of n points. Observe that this is not restrictive because X T is stored as a finite set of points in a computer.
2. Compute the connected components I i of r θ,y ∩Ŝ ε (X n ) according to the following steps: Initialize the list of the extremes of these intervals by listz= ∅ and listl= ∅. Then, for i = 1 to n: From the comments at the beginning of subsection 2.1, we know that, with probability one, listz equals r θ,y ∩ ∂Ŝ ε . • Sort listl. With probability one, listl has an even number, 2m, of elements (see the comments at the beginning of subsection 2.2), and define a i and b i such that 2(i−1)+1 = a i , 2i = b i (which corresponds to a i and b i in Definition 2.1; i.e. (a i , b i ) are the connected components of r θ,y ∩Ŝ ε (X)).
3 Obtain the a i and b i such that I i = (a i , b i ) are the connected components ofŜ 4ε (X n ) ∩ r θ,y by using the same procedure.
3 The approach based on the α-convex hull 3.1 The estimator based on the α -hull assuming the α-rolling ball condition In [5], it was proven that in dimension two, under some regularity assumptions, the length of the boundary of the α-shape of an iid sample converges to the length of the boundary of the set. The α-shape has the very good property that its boundary is very easy to compute, and hence so is its surface measure. Unfortunately, we are unsure that the results can be extended to higher dimensions. Nevertheless, considering the α-convex hull (which is quite close to the α-shape) allows us to extend the results on the surface measure to any dimension. The following deterministic theorem states that, for all 0 < α < α, the surface measure of the boundary of the α -convex hull X n ⊂ S converges to |∂S| d−1 with a rate that depends on d H (∂C α (X n ), ∂S).
Let α < α be a positive constant and let X n ⊂ S be a finite set such that d H (X n , S) < 1 2 αα α+α and Then, is one to one, and (1)).
As previously, we can deduce the convergence rates from the deterministic theorem and results in [5] under the iid assumption.
Then, with probability one, for n large enough, .
In this case we do not need the additional hypothesis of (C, ε 0 )-regularity. The convergence rate is far better than the one given in Theorem 2.2, where the price to pay is the computational cost when d increases. Indeed, as detailed in next section, the computation of the α-convex hull requires us to start by the computation of the Delaunay complex. With regard to the parameter selection α , a fully data driven (but computationally expensive) method is proposed in [38].

Computation with the use of Crofton's formula
Unfortunately, the explicit computation of |∂C α (X n )| d−1 is very difficult. However, from the results in Lemma 7.7, we derive that we can make use Crofton's formulae and the Monte Carlo method to estimate |∂C α (X n )| d−1 . This, as we will see, is based on the fact that the computation ofň α (θ, y) := n ∂Cα(Xn) (θ, y) is feasible. It first requires the computation of the α-convex hull, as well as the convex hull, of X n . Recall that the convex hull H(X n ) of X n is equal to the intersection of a finite number of half-spaces H(  In [24], it is proved for dimension 2 that C α (X n ) c is the union of a finite number of balls and the aforementioned half-spaces but mentioned that the generalization is not difficult. The centres O i of these balls and their radii r i are obtained by computing the Delaunay complex. The computational cost of the Delaunay complex is the main part of the computational cost of our algorithm, which is defined as follows: 1. Compute all the Delaunay simplices   Also compute Ω i (resp. ρ i ), which is the center (resp. radius) of the sphere circumscribed to X i 1 , . . . , X i d in the plane spanned by X i 1 , . . . , X i d .

Now we have two different cases:
(a) f i is a face of ∂H(X n ); that is, there exists j such that f i ⊂ H j .
Then, define w i = u j .
(b) f i is not a face of ∂H(X n ), thus there exists j ≤ K such that Figure 12: The convex hull of the points, all the B + i , the boundary faces (green) and two B − . B − 1 correspond to case (a) and B − 2 corresponds to case (b).
To simplify notation, we write C α (X n ) c = i B i . Observe that if the line r θ,y is chosen at random (w.r.t. dµ d−1 dθ), with probability one, then we have r θ,y ∩ ∂B i , which contains less than three points.
• For all z ∈ r θ,y ∩ ∂B i , if for all j z / ∈B j , then do list=list∪{z}.

Discussion of the rates of convergence
In Corollary 4, we obtained the same convergence rate as the one provided in [5] for d = 2, which is conjectured as suboptimal. As mentioned in [5], if the measure of the symmetric difference between S and an estimatorŜ n is bounded by ε n , then we can only expect that plug-in methods allow us to estimate |∂S| d−1 with a convergence rate ε n . Thus, in the iid setting, the estimator defined by (6) (respectively (7) to (9)) can be seen as 'optimal' relative to the use of the Devroye-Wise support estimator (respectively, the α-convex hull support estimator) because they achieve the best possible convergence rates for those estimators. This is nevertheless far from being optimal: the minimax rate is conjectured to be n − d+3 2d+2 , which is the minimax rate for the volume estimation problem (see [6]), and in [34] it is proved that the minimax rate is the same for the volume estimation problem and the surface area estimation problem (at least in the image setting, which usually extends to the iid setting). Unfortunately, attaining this optimal rate for the surface area estimation problem is much more involved, even in the easier setting with data uniformly drawn in S and S c with perfect identification. No estimator attaining this rate has yet been proposed.

Integralgeometric estimations via a Monte Carlo method and numerical experiments
To estimate the surface area with a Monte Carlo method, we propose the following classical procedure. Generate a random sample θ 1 , . . . , θ k that is uniformly distributed on (S + ) d−1 . For each i = 1, . . . , k, draw a random ..,n ||X j ||. Then, the estimators are given bŷ min(n εn,Xn (θ i , y i j ), N 0 ) (9) 6 Simulation study The performance of (8) and (10) is illustrated through a simulation study. We consider the sets S(d, r) = B d (O, 1) \B d (O, r) for d = 2, 3, r = 0.5, 0.6, 0.7, 0.8 and 0.9. On each set, we draw n = 50, 100, 200, 500, 1000, 2000 and 4000 iid random vectors supported on S(d, r) , whose common distribution is X = RZ, where R is a real valued random variable uniformly distributed on [1 − r, 1] and Z is a random vector (independent of R) that is supported on the (d − 1)-dimensional sphere.
For (8), we computed the parameter ε n as follows: for each sample point we calculate the distance to its closest point in the sample, and we choose ε n as the third quantile of these n distances. For (10), we estimated the parameter α with the data-driven estimator proposed in [38]. Roughly speaking, "the largest value of α compatible with the α-convexity assumption" is chosen.
To illustrate the convergence without the bias of the Monte Carlo step we compare our estimator with the Crofton based surface area estimation on the true (unknown) set based on the same line sample. More precisely, for each example (given by a dimension d, a radius r, a sample size n and an experiment number i) we draw X n as previously explained and 4000 values of (θ j , y j ) (i.e 4000 lines) and then compute : j=1 (n ε,Xn (θ j , y j ) − n ∂Sr (θ j , y j )) 4000 j=1 n ∂Sr (θ j , y j ) (11) and In Figure 13 we show, for each d and r, the results of the proposed method based on 57 experiment replications. Black curves represent results for r-convex hull based surface area estimator. We present here the evolution of the extremal values of the error given by (12) and (11) (dots), the 5% and 95% quantiles (dashed), the 25% , 50% and the 75% quantiles (plain). The convergence towards 0 (blue line) can be observed. In red we present the same curves for the case of the Devroye-Wise based estimator. As expected due to theoretical results, convergence is quicker for the r-convex hull estimator than for the Devroye-Wise based surface area estimator (same curves, in red). This is particularly clear when r 0 ≥ 0.7.

Sketch of the proofs of Theorems 2.2 and 2.3
The idea is to consider separately two subsets of the set of lines that intersect ∂Ŝ εn (X n ): 1. If a line r θ,y = y + Rθ is 'far enough' (fulfilling condition L(ε) for some ε > 0, see Definition 7.1) from the tangent spaces, then our algorithm allows a perfect estimation of n ∂S (θ, y), see Lemma 7.5.
2. Considering the set of lines that are not 'far enough' from the tangent spaces (denoted by A εn (θ)), see Definition 7.1), Corollary 5 states that, under (C, 0 )-regularity, the integral ofn εn,Xn (θ, y) on A εn (θ) is bounded from above by C ε 1/2 n , where C is a positive constant. Theorem 2.3 states that the previous bound can be improved to C ε n , under (C, 0 )-regularity, if ∂S has a bounded number of linear intersections.

Condition L(ε)
We now define the two sets of lines to be tackled separately. The lines that are 'far' from an affine tangent space and the lines that are 'close to being tangent' to ∂S. More precisely, recall that the unit outer normal vector η x at x is well defined under the rolling ball hypothesis (see Remark 1). Now we define the collection of all the affine (d − 1)-dimensional tangent spaces.

Some useful lemmas
Lemma 7.2. Let S be a compact set fulfilling the outside and inside α-rolling conditions. Let r θ,y be a line that fulfills condition L(0) and r θ,y ∩ ∂S = ∅. Then, r θ,y intersects ∂S in a finite number of points.
Proof. Because S fulfills the outside and inside α-rolling conditions, Theorem 1 in [42] implies that for any x ∈ ∂S, the affine (d − 1)-dimensional tangent space T x ∂S exists. If r θ,y fulfills L(0), then r θ,y is not included in any hyper-plane tangent to S. Suppose that ∂S ∩ r θ,y is not finite. Then, by compactness, one can extract a subsequence t n ∈ ∂S ∩ r θ,y that converges to y ∈ ∂S ∩ r θ,y .
1. Because t n and y are in r θ,y , we have that, for all n, (t n −y )/||t n −y || = ±θ.
These two facts imply that θ ∈ T y ∂S, which contradicts the assumption that r θ,y is not included in any hyper-plane tangent to S. Lemma 7.3. Let S ⊂ R d be a compact set fulfilling the outside and inside αrolling conditions. Let ε > 0 be such that ε < α/4 and ν = 2[2ε(α − 2ε)] 1/2 . For any line r θ,y fulfilling condition L(ε) and r θ,y ∩ ∂S = ∅, we have that r θ,y meets ∂S at a finite number of points t 1 , . .
Proof. If a line fulfills condition L(ε), then it fulfills condition L(0). Consequently, the fact that r θ,y intersects ∂S in a finite number of points follows from Lemma 7.2. Let us denote by t 1 < · · · < t k the intersection of r θ,y with ∂S. Let us denote by η t i and η t i+1 the outer normal vectors at t i and t i+1 , respectively. We have two cases: the open interval (t i , t i+1 ) ⊂ S c or (t i , t i+1 ) ⊂ S. Let us consider the first case (the proof for the second one is similar).
Because (t i , t i+1 ) ⊂ S c and S fulfills the inside α-rolling condition on t i , there exists a z ∈ S such that t i ∈ ∂B(z, α) and B(z, α) ⊂ S. In particular, Reasoning in the same way but with t i+1 , we get η t i+1 θ ≤ 0. Given that r θ,y is not included in any tangent hyperplane, we have that η t i , θ > 0 and η t i+1 , θ < 0.
Lemma 7.4. Let S ⊂ R d be a compact set fulfilling the outside and inside α-rolling conditions, with a (C, ε 0 )-regular boundary. Then, for all ε ≤ min{ε 0 , α}/4, Moreover, if ∂S has bounded number of linear intersections, then Proof. Observe that According to Remark 3 if y ∈ θ ⊥ : d(y, F θ ) = , then, for all t ∈ (0, ), r θ,y fulfills L( /4 − t). From the proof of the previous lemma, it follows that for any y ∈ θ ⊥ with d(y, F θ ) = and < 4ε, and any t ∈ (0, /4) Hence, with t → 0 we obtain n ∂S (θ, y) ≤ diam(S)( ) −1/2 /(2 √ α), from which: By the definition of ϕ θ , From the (C, ε 0 )-regularity of ∂S and the mean value theorem we obtain By applying exactly the same reasoning, under the hypothesis of the boundedness of the number of linear intersections for ∂S, we get Remark 4. If in the proof of Lemma 7.4 we take = 0, then we obtain that the measure of the set of lines belonging to some half-space tangent to ∂S is 0.
Lemma 7.5. Let S be a compact set fulfilling the outside and inside αrolling conditions. Let X n = {X 1 , . . . , X n } ⊂ S. Let ε n → 0 be such that d H (X n , S) ≤ ε n . Let r θ,y = y + Rθ be any line fulfilling condition L(ε n ). Then, for n large enough so that 4ε n < α, n ∂S (θ, y) =n εn,Xn (θ, y).
If r θ,y ∩ ∂S = ∅, then inequality (14) holds. Assume r θ,y ∩ ∂S = ∅. We will now prove that Because S fulfills the inside α-rolling condition on t i , there exists a z i ∈ S such that t i ∈ ∂B(z i , α) and . Reasoning in the same way but with t i+1 , η t i+1 , θ ≤ 0. By condition L(ε n ), we obtain η t i , θ > 0 and η t i+1 , θ < 0.
Suppose that for all t ∈ (t i , t i+1 ) we have d(t, ∂S) ≤ 4ε n . Take n large enough so that 4ε n < α. Because S fulfills the outside and inside α-rolling conditions, by Lemma 2.3 in [33], ∂S has positive reach greater than α. Then, by Theorem 4.8 in [25], γ = {γ(t) = π ∂S (t), t ∈ (t i , t i + 1)}, the orthogonal projection onto ∂S of the interval (t i , t i+1 ) is well defined and is a continuous curve in ∂S. By Theorem 1 in [42], the map from ∂S to , which, together with (16), ensures the existence of an s ∈ (t i , t i+1 ) such that d(s, γ(s)) ≤ 4ε n and θ ∈ η ⊥ γ(s) , which contradicts the assumption that r θ,y fulfills condition L(ε n ). This proves (15).
From (15), we easily obtain (because s ∈ S c and X n ⊂ S) that To conclude (14) let us write, for i = 1, . . . , k, I i = [t 2(i−1)+1 , t 2i ] for the connected components of S ∩ r θ,y . Since d H (X n , S) < ε n , S ⊂Ŝ εn (X n ). Then, for i = 1, . . . , k, there exists a j such that I i ⊂ I j , I j being a connected component ofŜ εn ∩ r θ,y . Note now that (17) ensures that, for all i = i , if I i ⊂ I j and I i ⊂ I j then I j and I j are not in the same connected component ofŜ 4εn (X n ) thus they are not glued, and thenn εn,Xn (θ, y) ≥ n ∂S (θ, y). Next, we will prove the opposite inequality, n εn,Xn (θ, y) ≤ n ∂S (θ, y).
Lemma 7.6. Let S ⊂ R d be a compact set fulfilling the outside and inside α-rolling conditions. Let X n ⊂ S and suppose ε n → 0 is a sequence such that d H (X n , S) ≤ ε n , while r θ,y = y + Rθ and A 1 , . . . , A k are the sets in Definition 2.1, A i = (a i , b i ) for i = 1, . . . , k. Now suppose that the sets are indexed in such a way that a 1 < b 1 < a 2 < . . . < b k . Then, for all i = 2, . . . , k, we have that ||a i − b i−1 || > 3 √ ε n α and for all i = 1, . . . , k, ||b i − a i || > 3 √ ε n α, for n large enough so that 3 √ αε n < α/2, which implieŝ Proof. Assume by contradiction that for some i, Moreover, π S is a continuous function.
, and the maximum is attained at some x 0 ∈ [b i−1 , a i ]. First, we show that ||x 0 − π S (x 0 )|| ≥ 3ε n . Indeed, suppose by contradiction that for all t ∈ (b i−1 , a i ), d(t, ∂S) ≤ 3ε n . Then, d(t, X n ) ≤ 4ε n , which contradicts the definition of the points a i and b i . The fact that ||x 0 − π S (x 0 )|| ≥ 3ε n > d(a i , S) = d(b i−1 , S) guarantees that x 0 ∈ (b i−1 , a i ) and that η 0 , the outward unit normal vector to ∂S at π S (x 0 ), is normal to θ.
The proof that for all i = 1, . . . , k, ||b i − a i || > 3 √ ε n α follows the same ideas, we will give a sketch of the proof. Let b i > a i (recall that we ordered the points a 1 < b 1 < . . . < a k < b k ) be such that Proceeding as before, max x∈[a i ,b i ] ||x − π S (x)|| ≥ 3ε n and it is attained at some x 0 ∈ (a i , b i ). Let z 0 = π S (x 0 ) − η 0 α, with η 0 being the outward unit normal vector to ∂S at π S (x 0 ). Then r θ, √ ε n α, which is a contradiction.
Corollary 5. Let S ⊂ R d be a compact set fulfilling the outside and inside α-rolling conditions and with a (C, ε 0 )-regular boundary. For n large enough so that 3 √ αε n < min(α/2, ε 0 ), we have

Proof of Theorem 2.3
The proof of Theorem 2.3 is basically the same as the previous one. Since N 0 ≥ N S Lemma 7.5 ensures that, for all r y,θ not in A εn (θ), min(n(θ, y), N 0 ) = n ∂S (θ, y), for n large enough that 4ε n < α. Thus, we still have, for n large enough, Now, by applying (13) for the first part and a similar calculation for the second part, we get that for n large enough.

Proof of Corollary 3
By Corollary 1 in [15], we know that, with probability one, for T large . . , X tn } be a discretization of X T such that t i − t i−1 = T /n and t n = T . Put ε n = d H (X n , S), then ε n ≥ ε T . It is clear that, for a fixed T , ε n decreases to ε T as n → ∞. To emphasize the dependence on the set, we will writeÎ N 0 d−1 (∂S, X n ) for the estimator based on X n , andÎ N 0 d−1 (∂S, X T ) for the estimator based on X T (both defined using Definition 2.1). Then, by (20), to prove Corollary 3 it is enough to proveÎ N 0 d−1 (∂S, X n ) →Î N 0 d−1 (∂S, X T ) as n → ∞, for arbitrary fixed T . Fix θ and y. It is clear thatn(θ, y)(∂S, X n ) → n(θ, y)(∂S, X T ) as n → ∞, and so Corollary 3 follows by the dominated convergence theorem, using the fact that min{n(θ, y), N 0 } ≤ N 0 .

Proofs for the estimator based on the α-hull
Theorem 3.1 will be easily obtained from the two following geometric lemmas and Theorem 3 in [37].
Here, we need to introduce some new notation. If f is a function, then ∇ f (x) denotes its gradient and H f its Hessian matrix. Given two sets C, D ⊂ R d , we write C ≈ D if there exists an homeomorphism between C and D.
In what follows, M ⊂ R d will be a compact set, and C 2 a (d − 1)-dimensional manifold (with or without boundary). Then for all x in M , there exists an r x > 0 such that either The set of points satisfying condition i) constitute int(M ), while the set of points satisfying ii) constitute ∂M . We have that ∂M is a (d − 2)dimensional manifold without boundary and, as a consequence, If M is a manifold as before, and ∂M = ∅, we define for any compact set E ⊂ M (E is not necessarily a manifold) its interior int(E) = {x ∈ E : ∃r x such that for all r ≤ r x ,B(x, r) ∩ E ≈ B d−1 (0, 1)}. We have int(E) is a manifold (without boundary and, when is not empty int(E) has the same dimension as M ).
3. ∂C α (X n ) ≈ ∂S Proof. Let us prove first that there are no isolated points in ∂C α (X n ). Indeed, suppose by contradiction that there exists x is an isolated point of ∂C α (X n ); that is, there exists r > 0 such that B(x, r) ∩ ∂C α (X n ) = {x}. By connectedness of B(x, r) \ {x} we have either B(x, r) \ {x} ⊂ C α (X n ) c or B(x, r) \ {x} ⊂ C α (X n ). The second case contradicts x ∈ ∂C α (X n ) because C α (X n ) is a close set. Thus, we have B(x, r) \ {x} ⊂ C α (X n ) c . Let us introduce x * = π ∂S (x), then ||x − x * || ≤ ε n . Let us denote η * = η x * . Since We will prove that α + α − ||O y − O|| ≥ d H (X n , S), which is a contradiction.
Because x ∈ C α (X n ) we have ||O y − x|| ≥ α . Let us write O y = y+aη * +bw with ||w|| = 1 and w ∈ (η * ) ⊥ , by ||O y −x|| ≥ α and ||O y −y|| < α it quickly comes that a 2 + b 2 ≤ (α ) 2 and a ≤ min(r,εn) Thus, ||O y − O|| is bounded from above by As announced, this leads to a contradiction. From (7) it follows that, for some N , Here, the B i are balls of radius r i larger than α or half-spaces (by abuse of notation, if B i is an half-space we will put r i = +∞). Our first step consists in proving that: Hence, d(Ω i , X n ) > α , and by continuity, there exists a t > 0 so small that d( Second, consider the case where u i is a unit vector. We can conclude, similarly, on introducing Ω i = x + α u i , that B(Ω i , α ) ∩ X n = ∅ and B(Ω i − tu i , α ) ⊂ C α (X n ) c (for some positive but small enough t) and so x ∈C α (X n ) c .
If x ∈ ∂C α (X n ) ∩ X n , then by the preliminary result, there exists a sequence (x k ) in ∂C α (X n ) \ {x} with x k → x. Because X n is finite, it follows that for k large enough, x k ∈ ∂C α (X n )∩X c n . Because the number of possible S i is finite, we can extract from (x k ) a sequence (x k ) such that there exists a S i = ∂B i such that for all k, x k ∈ S i making k → +∞, and then we have x ∈ S i .

Our second step consists in proving that if there exists an
α and x * = π ∂S (x). Observe that from the first step we know that B i =B(O i , α ). Write η x * for the outward (from S) unit normal vector of ∂S at x * and O * = x * − αη x * .
Note first thatB Introduce Figure  15). Then, from the second inclusion in (22), we get y ∈ S, and from the first inclusion in (22) we get d(y, C α (X n )) ≥ ||y − y * ||. Then, ||y − y * || ≤ ε n , which in turn implies where in the first inequality of the last line we bounded , and in the last inequality α − ≥ α/2, thus, combined with Equation (23), we can conclude the proof of Equation (21).
As the third step, we will now conclude the proof of assertion 1. Note that if B i is a ball (and not an half-space), then ∂B i ∩ B c j = ∂B i ∩ P i,j where P i,j the following closed half space.
Put C i (X n ) = (∂B i ∩H i )\X n . We are going to prove that C i (X n ) satisfies conditions (a), (b), (c) and (d) of assertion 1. First note that (a) is obvious by construction. Suppose x ∈ ∂C α (X n ) \ X n . By the first step, we know that there exists a B i 0 which is a ball of radius α such that x ∈ S i 0 and thus we are in the situation where x ∈ ∂B i 0 ∩ H i 0 with H i 0 a convex polygon. If now x ∈ ∂C α (X n ) \ X n but x / ∈ ∪C i (X n ), then we must have x ∈ ∂B i 0 ∩ ∂H i 0 . This gives and thus |∂C α (X n ) \ i C i (X n ) | d−1 = 0, which proves (b). We will now prove that if i = j and B i and B j are two balls, then (∂B i ∩ H i )∩(∂B j ∩H j ) = ∅. Suppose by contradiction that (S i ∩H i )∩(S j ∩H j ) = ∅, then ||x − O i || 2 − r 2 i < ||x − O j || 2 − r 2 j and ||x − O i || 2 − r 2 i ≥ ||x − O j || 2 − r 2 j , which is a contradiction. Thus, if C i (X n ) and C j (X n ) are both non-empty, then we have that B i and B j are two balls, and if i = j, C i (X n )∩C j (X n ) = ∅, which proves (d).
This also proves that if x ∈ C i (X n ), then there exists an r x > 0 small enough so that ∂C α (X n ) ∩ B(x, r x ) = ∂B i ∩ B(x, r x ). Thus, ∂C α (X n ) ∩ B(x, r x ) is a C 2 , (d − 1)-dimensional manifold. Moreover, the tangent space at x is given by (x − O i ) ⊥ . Also, the unit normal (to ∂C α (X n )) vector (O i − x)/||x − O i || is well defined, and points outwards to C α (X n ). This concludes the proof of (c) and also the proof of 1).
The proof of 2) follows the same ideas used to prove Theorem 3 in [1]. We are going to give the main steps of the proof (adapted to our case).
Finally, we prove 3. Since reach(∂S) ≥ α and d H (∂C α (X n ), ∂S) ≤ ε n < α, π ∂S , restricted to ∂C α (X n ), is continuous (see [25]). The continuity of π −1 ∂S : ∂S → ∂C α (X n ) follows from the same ideas used to prove the injectivity of π ∂S : we provide a sketch of the proof. It follows from reach(∂S) ≥ α that π −1 ∂S (x) = x − (x)η x with (x) ≥ 0. In addition, x → η x is a continuous function (see Theorem 1 in [42]). It remains to be proved that is a continuous function. If this is not the case, then we can find sequences (y n ) ⊂ ∂S and (y n ) ⊂ ∂S, both converging to some y ∈ ∂S), such that (y n ) → 1 and (y n ) → 2 . We can conclude exactly as in the proof of injectivity that we can take x 1,n = y n − (y n )η yn and x 2,n = y n − (y n )η y n making n → +∞. We thus have ∂S ≈ ∂C α (X n ), which proves assertion 3, and thus concludes the proof of the lemma. Lemma 7.8. Suppose that M is a C 2 , bounded (d−1)-dimensional manifold with positive reach α. Let π M denote the projection onto M andM be a C 2 , (d − 1)-dimensional manifold such that 1. π M is one to one fromM to M ,    (24) .