Brownian limits, local limits and variance asymptotics for convex hulls in the ball
Schreiber and Yukich [Ann. Probab. 36 (2008) 363–396] establish an asymptotic representation for random convex polytope geometry in the unit ball , in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the -face and intrinsic volume functionals.
10.1214/11-AOP707 \volume41 \issue1 2013 \firstpage50 \lastpage108 \newproclaimdefnDefinition[section] \newproclaimremaRemark \newproclaimremarkRemark \newproclaimremarksRemarks
Convex hulls in the ball
A]\fnmsPierre \snmCalkalabel=e1]email@example.com, B]\fnmsTomasz \snmSchreiber\thanksreft1,au2 and C]\fnmsJ. E. \snmYukich\corref\thanksrefau3label=e3]firstname.lastname@example.org \dedicatedDedicated to the memory of Tomasz Schreiber \thankstextt1Born June 25, 1975; died on December 1, 2010. \thankstextau2Supported in part by the Polish Minister of Science and Higher Education Grant N N201 385234 (2008–2010). \thankstextau3Supported in part by NSF Grant DMS-08-05570.
class=AMS] \kwd[Primary ]60F05 \kwd[; secondary ]60D05. Functionals of random convex hulls \kwdparaboloid growth and hull processes \kwdBrownian sheets \kwdstabilization.
Let be a smooth convex set in of unit volume. Letting be a Poisson point process in of intensity , we let be the convex hull of . The random polytope , together with the analogous polytope , obtained by considering i.i.d. uniformly distributed points in , are well-studied objects in stochastic geometry.
The study of the asymptotic behavior of the polytopes and , as and , respectively, has a long history originating with the work of Rényi and Sulanke RS (). Letting denote the unit sphere, the following functionals of have featured prominently:
the volume of abbreviated as ;
the number of -dimensional faces of , denoted ; in particular is the number of vertices of
the mean width of ;
the distance between and in the direction , here denoted ;
the distance between the boundary of the Voronoi flower, defined by and , in the direction , here denoted ;
the th intrinsic volumes of here denoted .
The mean values of these functionals on general convex polytopes, as well as their counterparts for , have been widely studied, and for a complete account we refer to the surveys of Affentranger Af (), Buchta Bu1 (), Gruber Gr (), Reitzner ReBook (), Schneider Sc1 (), Sc2 () and Weil and Wieacker WW (), together with Chapter 8.2 in Schneider and Weil SW (). There has been recent progress in establishing higher order and asymptotic normality results for these functionals, for various choices of . We signal the important breakthroughs by Reitzner Re (), Bárány and Reitzner BR2 (), Bárány et al. BFV (), Pardon Pa () and Vu VV05 (), VV (). These results, together with those of Schreiber and Yukich SY (), are difficult and technical, with proofs relying upon tools from convex geometry and probability, including martingales, concentration inequalities and Stein’s method. When is the unit radius -dimensional ball centered at the origin, Schreiber and Yukich SY () establish variance asymptotics for as , but up to now little is known regarding explicit variance asymptotics for other functionals of .
This paper has the following goals. We first study two processes in formal space–time , one termed the paraboloid growth process and denoted by , and a second termed the paraboloid hull process, denoted by . While the first process was introduced in SY (), the second has apparently not been considered before. When , an embedding of convex sets into the space of continuous functions on , together with a re-scaling, show that these processes are naturally suited to the study of . Their spatial localization can be exploited to describe first and second order asymptotics of functionals of . Many of our main results, described as follows, are obtained via geometric properties of the processes and . Our goals are as follows:
Show that the distance between and , upon re-scaling in a local regime, converges in law as , to a continuous path stochastic process defined in terms of , adding to Molchanov Mo (); similarly, we show that the distance between and the Voronoi flower defined by converges in law to a continuous path stochastic process defined in terms of . In the two-dimensional case the fidis (finite-dimensional distributions) of these distances, when re-scaled, are shown to converge to the fidis of and , whose description is given explicitly, adding to work of Hsing hsing ().
Show, upon re-scaling in a global regime, that the suitably integrated local defect width and defect volume functionals, when considered as processes indexed by points in mapped on via the exponential map, satisfy a functional central limit theorem, that is, converge in the space of continuous functions on to a Brownian sheet on the injectivity region of the exponential map, whose respective variance coefficients and are expressed in closed form in terms of and To the best of our knowledge, this connection between the geometry of random polytopes and Brownian sheets is new. In particular we show
This adds to Reitzner’s central limit theorem (Theorem 1 of Re ()) and his variance approximation (Theorem 3 and Lemma 1 of Re ()), both valid when is an arbitrary smooth convex set. It also adds to Hsing hsing (), which is confined to the case .
Establish central limit theorems and variance asymptotics for the number of -dimensional faces of , showing for all ,
Establish central limit theorems and variance asymptotics for the intrinsic volumes establishing for all that
where again is described in terms of the processes and . This adds to Bárány et al. (Theorem 1 of BFV ()), which shows .
Limits (1)–(4) resolve the issue of finding variance asymptotics for face functionals and intrinsic volumes, a long-standing problem put forth this way in the 1993 survey of Weil and Wieacker (page 1431 of WW ()): “We finally emphasize that the results described so far give mean values hence first-order information on random sets and point processes… There are also some less geometric methods to obtain higher-order informations or distributions, but generally the determination of the variance, for example, is a major open problem.”
These goals are stated in relatively simple terms, and yet they and the methods behind them suggest further objectives involving additional explanation. One of our chief objectives is to carefully define the growth processes and and exhibit their geometric properties making them relevant to , including their localization in space, known as stabilization. The latter property is central to establishing variance asymptotics and the limit theory of functionals of . A second objective is to describe two natural scaling regimes, one suited for locally defined functionals of , and the other suited for the integrated characteristics of , namely the width and volume functionals. A third objective is to extend the afore-mentioned results to ones holding on the level of measures. In other words, functionals considered here are naturally associated with random measures, and we shall show variance asymptotics for such measures and also convergence of their fidis to those of a Gaussian process under suitable global scaling. We originally intended to restrict attention to convex hulls generated from Poisson points with intensity density , but realized that the methods easily extend to treat intensity densities decaying as a power of the distance to the boundary of the unit ball as given by (5) below, and so we shall include this more general case without further complication. These major objectives are discussed further in the next section.
The extension of the variance asymptotics (2) and (3) to smooth compact convex sets with a boundary of positive Gaussian curvature is nontrivial and is addressed in CY (). We expect that much of the limit theory described here can be “de-Poissonized,” that is to say, extends to functionals of the polytope . This extension involves challenging technical questions which we do not address here.
2 Basic functionals and their scaled versions
Given a locally finite subset of , we denote by the convex hull generated by For a given compact convex set containing the origin, we let be the support function of that is to say, for all , we let . It is easily seen for and that
For the radius-vector function of in the direction of is given by
For and we abuse notation and henceforth denote by the Poisson point process in of intensity
The parameter shall remain fixed throughout, and therefore we suppress mention of it. Further, abusing notation we put
The principal characteristics of studied here are the following functionals, the first two of which represent in terms of continuous functions on :
The defect support function. For all , we define
where for we define In other words, is the defect support function of in the direction It is easily verified that is the distance in the direction between the sphere and the Voronoi flower
where for and we let denote the -dimensional radius ball centered at .
The defect radius-vector function. For all , we define
where for and we put Thus, is the distance in the direction between and The convex hull contains the origin, except on a set of exponentially small probability as , and thus for asymptotic purposes we assume without loss of generality that always contains the origin, and therefore the radius vector function is well defined.
The numbers of -faces. Let , , be the number of -dimensional faces of In particular, and are the number of vertices and edges, respectively. The spatial distribution of -faces is captured by the -face empirical measure (point process) on given by
Here is the collection of all -faces of and , , is the point of which is closest to with ties ignored as they occur with probability zero (there are other conceivable choices for , but we find this one to be as good as any). The total mass coincides with
Projection avoidance functionals. Representing intrinsic volumes of as the total masses of the corresponding curvature measures, while suitable in the local scaling regime, turns out to be less useful in the global scaling regime, as it leads to an asymptotically vanishing add-one cost for related stabilizing functionals, thus precluding normal use of stabilization theory. To overcome this problem, we shall use the following consequence of Crofton’s general formula, usually going under the name of Kubota’s formula; see (5.8) and (6.11) in SW (). We write
where is the th Grassmannian of , is the normalized Haar measure on and is the orthogonal projection of onto the -dimensional linear space We shall only focus on the case because for , we have for all nonempty, compact convex ; see page 601 in SW (). Write
where Putting this yields
Noting that and interchanging the order of integration, we conclude, in view of the discussion on pages 590–591 of SW (), that the considered expression equals
where is the -dimensional linear space spanned by , is the set of -dimensional linear subspaces of containing and is the corresponding normalized Haar measure; see SW (). Thus, putting
and using (10), we are led to
We will refer to as the projection avoidance function for .
The large asymptotics of the above characteristics of are studied in two natural scaling regimes, the local and the global one, as discussed below.
Local scaling regime and locally re-scaled functionals. The first scaling we consider is referred to as the local scaling in the sequel. It stems from the following observation, which, while considered before in BR2 (), shall be discussed here in the context of stabilization of growth processes. If we consider the local behavior of functionals of in the vicinity of two fixed boundary points with , then these behaviors become asymptotically independent. Moreover, if approaches slowly enough as the asymptotic independence is preserved. On the other hand, if the distance between and decays rapidly enough, then both behaviors coincide for large , and the resulting picture is rather uninteresting. As in SY (), it is therefore natural to ask for the frontier of these two asymptotic regimes and to expect that this corresponds to the natural characteristic scale between the observation directions and where the crucial features of the local behavior of are revealed.
To render the characteristic scale as transparent as possible, we start with some simple yet important observations, which shall eventually lead to asymptotic independence of local convex hull geometries and which shall also suggest the proper scaling limits of convex hull statistics. For arbitrary points , the support function of the convex hull satisfies for all , the relation
We make the fundamental observation that the epigraph of is thus the union of epigraphs which, locally near the apices, are of parabolic structure. Any scaling transformation for on the characteristic scale must preserve this structure, as should the scaling limit for .
To determine the proper local scaling for our model, we consider the following intuitive argument. To obtain a nontrivial limit behavior we should re-scale in a neighborhood of , both in the surfacial (tangential) directions with factor and radial direction with factor with suitable scaling exponents and so that:
The re-scaling compensates the intensity of with growth factor . In other words, a subset of in the vicinity of , having a unit volume scaling image, should host on average points of the point process Since the integral of the intensity density (5) scales as , with respect to the tangential directions, and since it scales as with respect to the radial direction, where we take into account the integration over the radial coordinate, we are led to and thus
The local behavior of the convex hull close to the boundary of , as described by the locally parabolic structure of , should preserve parabolic epigraphs, implying for that , and thus
We next describe scaling transformations for Fix , and let denote the tangent space to at . The exponential map maps a vector of the tangent space to the point , such that lies at the end of the geodesic of length starting at and having direction Note that is geodesically complete in that the exponential map is well defined on the whole tangent space although it is injective only on Instead of , we shall write or simply , and we make the default choice We use the isomorphism without further mention, and we shall denote the closure of the injectivity region of the exponential map simply by Thus we have
Further, consider the following scaling transform mapping into
Here is the inverse exponential map, which is well defined on and which takes values in the injectivity region . For formal completeness, on the “missing” point , we let admit an arbitrary value, say and likewise we put where denotes either the origin of or , according to the context. It is easily seen that is a.e. (with respect to Lebesgue measure on ) a bijection from onto the -dimensional solid cylinders
Throughout points in are written as , and we represent generic points in by , whereas we write to represent points in the scaled region . We assert that the transformation , defined at (16), maps the Poisson point process to , where is the dilated Poisson point process in the region having intensity
at Indeed, this intensity measure is the image by the transformation of the measure on given by
Next, notice that the exponential map has the following expression:
with . Therefore, since , we have
Since , this gives
We also have that
In Section 4, following SY (), we shall embed into a space of paraboloid growth processes on . One such process, denoted by and defined at (54), is a generalized growth process with overlap whereas the second, a dual process denoted by and defined at (61) is termed the paraboloid hull process. Infinite volume counterparts to and , described fully in Section 3 and denoted by and , respectively, play a natural role in describing the asymptotic behavior of our basic functionals of interest, re-scaled as follows:
The re-scaled version of the projection avoidance function (11) defined by
Global scaling regime and globally re-scaled functionals. The asymptotic independence of local convex hull geometries at distinct points of as discussed above, suggests that the global behavior of both and is, in large asymptotics, that of the white noise. Therefore it is natural to consider the corresponding integral characteristics of and to ask whether, under proper scaling, they converge in law to a Brownian sheet. Define the processes
where the “segment” for is the rectangular solid in with vertices and that is to say, , with standing for the th coordinate of . We shall also consider the cumulative values
Notice that the radius-vector function of the Voronoi flower coincides with the support function of . In particular, the volume outside is equal to
Since goes to uniformly, the volume outside is asymptotically equivalent to the integral of the defect support function, which in turn is proportional to the defect mean width by Cauchy’s formula. Moreover, in two dimensions the mean width is the ratio of the perimeter to (see page 210 of Schn ()), and so coincides with minus the mean width of , and consequently itself equals minus the perimeter of for On the other hand, is asymptotic to the volume of whence the notation for (asymptotic) width and for (asymptotic) volume.
To get the desired convergence to a Brownian sheet, we put
we show in Section 8 that it is natural to re-scale the processes and by and that the resulting re-scaled processes
converge in law to a Brownian sheet with an explicit variance coefficient.
Putting the picture together. The remainder of this paper is organized as follows.
Section 3. Though the formulation of our results might suggest otherwise, there are crucial connections between the local and global scaling regimes. These regimes are linked by stabilization and the objective method, which together show that the behavior of locally defined processes on the finite volume rectangular solids , defined at (17), can be well approximated by the local behavior of a related “candidate object,” either a generalized growth process or a dual paraboloid hull process , on an infinite volume half-space. While generalized growth processes were developed in SY () in a larger context, our limit theory depends heavily on a new object, the dual paraboloid hull process. The purpose of Section 3 is to carefully define these processes and to establish properties relevant to their asymptotic analysis.
Section 4. We show that as both and , defined, respectively, at (23) and (24), converge in law to continuous path stochastic processes explicitly constructed in terms of the paraboloid generalized growth process and the paraboloid hull process , respectively. This adds to Molchanov Mo (), who considers the “epiconvergence” in the space of the random process, arising as the binomial counterpart of . Molchanov’s results Mo () are not framed in terms of the rescaled function , and thus they do not involve the paraboloid growth processes described in this paper.
Section 5. When , after re-scaling in space by a factor of and in time (height coordinate) by , we use nonasymptotic direct considerations to provide explicit asymptotic expressions for the fidis of and as . These distributions coincide with the fidis of the parabolic growth process and the parabolic hull process , respectively.
Section 6. Both the paraboloid growth process and its dual paraboloid hull process are shown to enjoy a localization property, which expresses, in geometric terms, a type of spatial mixing. This provides a direct route toward establishing first and second order asymptotics for the convex hull functionals of interest.
Section 7. This section establishes closed form variance asymptotics for the total number of -faces as well as the intrinsic volumes for the random polytope . We also establish variance asymptotics and a central limit theorem for the properly scaled integrals of continuous test functions against the empirical measures associated with the functionals under proper scaling.
Section 8. Using the stabilization properties established in Section 6, we establish a functional central limit theorem for and showing that these processes converge, as in the space of continuous functions on , to Brownian sheets with variance coefficients given in terms of the processes and , respectively.
3 Paraboloid growth and hull processes
In this section we introduce the paraboloid growth and hull processes in the upper half-space often interpreted as formal space–time below, with standing for the spatial dimension and standing for the time dimension. Although this interpretation is purely formal in the convex hull set-up, it provides a link to a well-established theory of growth processes studied by means of stabilization theory; see below for further details. These processes turn out to be infinite volume counterparts to finite volume paraboloid growth processes, which are defined in the next section, and which are used to describe the behavior of our basic re-scaled functionals and measures.
Poisson point process on half-spaces. Fix , and let be a Poisson point process in with intensity density
In the sequel we shall show that the scaled Poisson point process with intensity defined at (18) converges to on compacts, but for now we use the process to define growth processes on half-spaces. As with and we suppress and simply write for .
Paraboloid growth processes on half-spaces. We introduce the paraboloid generalized growth process with overlap (paraboloid growth process for short), specializing to our present set-up the corresponding general concept defined in Section 1.1 of SY () and designed to constitute the asymptotic counterpart of the Voronoi flower Let be the epigraph of the standard paraboloid that is,
We introduce one of the fundamental objects of this paper.
Given a locally finite point set in , the paraboloid growth model is defined as the Boolean model with paraboloid grain and with germ collection , namely
where stands for Minkowski addition. In particular, we define the paraboloid growth process , where is the Poisson point process defined at (32).
The model arises as the union of upwards paraboloids with apices at the points of (see Figure 1), in close analogy to the Voronoi flower , where to each we attach a ball (which asymptotically scales to an upwards paraboloid as we shall see in the sequel) and take the union thereof.
The name generalized growth process with overlap comes from the original interpretation of this construction SY (), where stands for space–time with corresponding to the spatial coordinates and the semi-axis corresponding to the time (or height) coordinate, and where the grain possibly admitting more general shapes as well, arises as the graph of the growth of a germ born at the apex of and growing thereupon in time with properly varying speed. We say that the process admits overlaps because the growth does not stop when two grains overlap, unlike in traditional growth schemes. We shall often use this space–time interpretation and refer to the respective coordinate axes as to the spatial and time (height) axis.
The boundary of the random closed set constitutes a graph of a continuous function from (space) to (time), also denoted by in the sequel. In what follows we interpret , defined at (23), as the boundary of a growth process , defined at (54) below, on the finite region at (17); we shall see in Section 4 that is the scaling limit for the boundary of .
A germ point is called extreme in the paraboloid growth process if and only if its associated epigraph is not contained in the union of the paraboloid epigraphs generated by other germ points in that is to say,
For to be extreme, it is sufficient, but not necessary, that fails to be contained in paraboloid epigraphs of other germs. Write for the set of all extreme points.
Paraboloid hull process on half-spaces. The paraboloid hull process can be regarded as the dual to the paraboloid growth process. At the same time, the paraboloid hull process is designed to exhibit geometric properties analogous to those of convex polytopes with paraboloids playing the role of hyperplanes, with the spatial coordinates playing the role of spherical coordinates and with the height/time coordinate playing the role of the radial coordinate. The motivation of this construction is to mimic the convex geometry on second order paraboloid structures in order to describe the local second order geometry of convex polytopes, which dominates their limit behavior in smooth convex bodies. As we shall see, this intuition is indeed correct and results in a detailed description of the limit behavior of
To proceed with our definitions, we let be the downwards space–time paraboloid hypograph
The idea behind our interpretation of the paraboloid process is that the shifts of correspond to half-spaces not containing in the Euclidean space We shall argue the paraboloid convex sets have properties strongly analogous to those related to the usual concept of convexity. The corresponding proofs are not difficult and will be presented in enough detail to make our presentation self-contained, but it should be emphasized that alternatively the entire argument of this paragraph could be re-written in terms of the following trick. Considering the transform , we see that it maps translates of to half-spaces and thus whenever we make a statement below in terms of paraboloids and claim it is analogous to a standard statement of convex geometry, we can alternatively apply the above auxiliary transform, use the classical result and then transform back to our set-up. We do not choose this option here, finding it more aesthetic to work directly in the paraboloid set-up, but we indicate at this point the availability of this alternative.
The next definitions are central to the description of the paraboloid hull process. Recall that the affine hull is the set of all affine combinations
For any collection of points in with affinely independent spatial coordinates we define to be the hypograph in of the unique space–time paraboloid in the affine space with quadratic coefficient and passing through
In other words is the intersection of and a translate of having all on its boundary; while such translates are nonunique for their intersections with all coincide.
For the parabolic segment is the unique parabolic segment with quadratic coefficient joining to in More generally, for any collection of points in with affinely independent spatial coordinates, we define the paraboloid face by
Clearly, is the smallest set containing and with the paraboloid convexity property: For any two it contains, it also contains In these terms, is the paraboloid convex hull In particular, we readily derive the property
Next, we say that is upwards paraboloid convex (up-convex for short) if and only if:
for each two we have
and for each we have
Whereas the first condition in the definition above is quite intuitive, the second will be seen to correspond to our requirement that as gets transformed to upper infinity in the limit of our re-scalings. Indeed, though is not defined at , the last coordinate of goes to when , and goes to when .
With the notation introduced above, we now define the second fundamental object of this paper.