Central limit theorems for random polygons in an arbitrary convex set
Abstract
We study the probability distribution of the area and the number of vertices of random polygons in a convex set . The novel aspect of our approach is that it yields uniform estimates for all convex sets without imposing any regularity conditions on the boundary . Our main result is a central limit theorem for both the area and the number of vertices, settling a wellknown conjecture in the field. We also obtain asymptotic results relating the growth of the expectation and variance of these two functionals.
10.1214/10AOP568 \volume39 \issue3 2011 \firstpage881 \lastpage903 \newproclaimdefinition[theorem]Definition
Random polygons
A]\fnmsJohn \snmPardon\correflabel=e1]jpardon@princeton.edu
class=AMS] \kwd52A22 \kwd60D05 \kwd60F05. Random polygons \kwdcentral limit theorem.
1 Introduction
Consider a Poisson point process in a convex set of intensity equal to the Lebesgue measure. We denote by the convex hull of the points of this process; is called a random Poisson polygon. We denote by the number of vertices of and by the area of . In this paper, we develop techniques to study the distributions of these random variables. Our main result is a central limit theorem, which is uniform over the set of all convex :
Theorem 1.1
As , we have the following central limit theorems for :
(1)  
(2) 
Here where is the standard normal distribution.
The novel aspect of our approach is that we require no regularity on ; it is this that enables us to obtain bounds which are uniform over all convex sets. Previous results on random polygons analogous to Theorems 1.1 have been confined to two cases: (i) a polygon groen1 (), groen2 () and (ii) of class with nonvanishing curvature hsing (). The key part of our argument is our use of a new compactness result for various types of local configuration spaces of convex boundaries.
As a consequence of our techniques, we also prove the following:
Theorem 1.2
As , we have the following estimates for ^{1}^{1}1After this paper was written, we learned that Imre Bárány and Matthias Reitzner have independently proved this result, as well as the closely related Corollary 1.4.:
(3) 
In other words, there is (up to a constant factor) only one parameter, say , which controls the asymptotics of the distributions of and . Thus, for example, the error terms in Theorem 1.1 could have instead been stated in terms of the variances.
For completeness, we should mention what is known about the growth of (say) , which can be effectively estimated using elementary geometric and combinatorial techniques. In dimension two, one has
(4) 
[In particular, the error terms in Theorem 1.1 go to zero as .] The estimate (4) is a consequence of the economic cap covering lemma of Bárány and Larman baranycap () in combination with other estimates in baranycap () and those of Groemer groemersphere () (in fact, their results apply to higher dimensions as well). We remark that the lower asymptotic is achieved when is a polygon, and the upper asymptotic is achieved when is with nonvanishing curvature.
We conclude by remarking that in recent years there has been significant progress in the study of random polytopes, but again most results deal only with the cases when (i) is a polytope baranypolytope (), and (ii) is with nonvanishing Gauss curvature reitzner1 (), vu1 (). We believe that an approach similar to ours should be possible in higher dimensions as well. This would shed new light on problems in that setting, and ultimately show that there is no qualitative difference between the cases (i) and (ii).
1.1 The uniform model random polygons
A model related to is where are i.i.d. uniformly in ; is called a random polygon. This is often referred to as the “uniform model” whereas is the “Poisson model.” Morally they are the same process in the limit (though making this precise is often difficult). It has been a wellknown open problem to prove central limit theorems for functionals of . For instance, Van Vu vuopen () has asked the question of whether a central limit theorem holds for , though the problem is a very natural one in the study of random polygons, a subject that began with work of Rényi and Sulanke renyisulanke1 (), renyisulanke2 (). Theorems 1.1 and 1.2 both carry over to the setting of , thus answering this question in the affirmative.
Corollary 1.3
As , we have the following central limit theorems for :
(5)  
(6) 
uniformly over all convex . Here where is the standard normal distribution.
Corollary 1.4
As , we have the following estimates for :
(7) 
uniformly over all convex .
As in the case of the Poisson model, these results are well known in the field in the two cases (i) a polygon and (ii) of class with nonvanishing curvature. The innovation in this paper is that all are treated uniformly.
A detailed derivation of Corollaries 1.3 and 1.4 from Theorems 1.1 and 1.2 will appear elsewhere pardonsupplement (). Suffice it to say here that they are almost immediate consequences of the corresponding results on the Poisson model once one proves that when , the variables and [as well as and ] have the same expectation and variance up to a small enough error.
2 The basic decomposition
In this section, we illustrate our basic approach. We will aim for Theorem 1.1, and Theorem 1.2 will be a corollary of our methods.
First, we observe that the functionals and both enjoy decompositions into local pieces. We define to equal the number of edges of whose angle lies in the interval . The definition of is best explained graphically (see Figure 1). Thus for any fixed sequence of angles , we have the following decompositions:
(8)  
(9) 
During the proof, we often do not need to distinguish between whether we are dealing with or . Thus we will use to denote either or when a statement holds for both.
A central limit theorem will follow if we can find a choice of such that the moments of are bounded uniformly, and such that the dependence between and becomes small as . Our construction is to choose so that the intervals have constant affine invariant measure (a measure depending on ). In this paper, we give a more or less explicit description of the affine invariant measure, which in practice should allow its easy estimation for any given class of convex sets, and thus a complete description of the behavior of random Poisson polygons and random polygons. As we remarked in the Introduction, a key result is the compactness of various configuration spaces.
After fixing notation in Section 3, we define the affine invariant measure in Section 4. Section 5 is devoted to the crucial step of proving the compactness of the configuration spaces. Using the information coming from compactness:
The remainder of the paper contains the explicit deduction of Theorems 1.1 and 1.2.
3 Notation and definitions
In this paper, will always denote a (bounded) convex set in .
We warn the reader that in most of the literature, one fixes and then considers a Poisson process of intensity . We have chosen instead to use the normalization and let . This is convenient for us because it makes many of our formulas simpler to state.
Any constants implied by the symbols , or are absolute; in particular they are not allowed to depend on . There will be times when we require ; this is no real restriction to us since in the end we will take . The group is the group of (oriented) area preserving affine transformations of ; it acts naturally on the entire problem studied here.
Many of the following definitions are illustrated in Figure 2. We may leave out the subscript later when doing so is unambiguous. {definition} We define the random variable to be the vertex of which has an oriented tangent line at angle . This is illustrated in Figure 2(a). {definition} A cap at angle is the intersection of with a halfplane at angle . We may specify a cap at angle by giving either its area or a point . These are denoted and , respectively; the latter is illustrated in Figure 2(b). {definition} We define the real number to be the area of the cap .
Lemma 3.1
The random variable has probability distribution given by where is the Lebesgue measure.
This follows directly from the definition of a Poisson point process. {definition} We define the function as follows:
(10) 
It will be important to have the following bound on the growth of :
Lemma 3.2
If , then
(11) 
The bound above is sharp; for instance for (i.e., the first quadrant).
Proof of Lemma 3.2 Project along the lines at angle to get a height function ; in Figure 3, is the length of the thick segment.
Now if then . Thus we see that it suffices to show that the function
(12) 
is decreasing. Differentiating with respect to , we see that it suffices to show that
(13) 
For , the lefthand side is clearly nonnegative, and the derivative of the lefthand side equals , which is by concavity of .
Lemma 3.3
If , then for .
Refer to Figure 4. The area of the upper trapezoid is
since it is contained in . The area of the lower triangle is since it contains and . Similar triangles gives the following inequality:
(14) 
Simplifying yields .
4 The affine invariant measure
Proposition 4.1
For every , we have
(15) 
where is the action of on line slopes. We say “ is affine invariant.”
Define to be the vector of length parallel to the chord whose length gives . Then we have
(16) 
The righthand side is invariant under the action of , so the result follows. {definition} We define the affine invariant measure to be .
The wet part of is defined as the union of all caps of area . In the literature, estimates for random polygons are frequently expressed in terms of the area of the wet part of . It is, perhaps, not surprising that our notion of the affine invariant measure is related to the area of the wet part in the following manner:
Lemma 4.2
One has the following relation:
(17) 
Consider the area swept out by the line segments bounding the caps of area at angles (area covered twice is counted twice). On the one hand, this area just equals
(18) 
On the other hand, we may express the area as an integral . Each line segment rotates about its midpoint (since the area of the caps is constant), so the area covered is just the integral of . Comparing this with (18) yields the result.
5 Compactness of configuration spaces
{definition}Define a configuration space for as follows. The objects of are convex subsets of of area with a distinguished line segment on their boundary. As a set, is equal to everything of the form , where is any convex set of area and is a halfplane such that has area . A typical member of is illustrated in Figure 5(a). We emphasize that the space does not depend on any choice of convex set ; rather it is the space of all caps of area that come from some convex set of area .
We call the configuration space of caps of area . If , then we call the distinguished part of its boundary its flat boundary and the undistinguished part of its boundary its convex boundary. We let the halfplane of equal the unique halfplane which contains and whose boundary contains the flat boundary of (this is exactly the appearing above).
We topologize by using the Hausdorff metric to compare both the set and its distinguished subset. Explicitly, . Let us observe that there is a natural action of on ; it is continuous. Certainly is not compact, since the group is noncompact. However, we will show directly that is compact. This simple fact will be an essential tool in virtually all of the estimates in the remainder of this paper.
Lemma 5.1
The space is compact.
Let be a sequence of elements of . Pick representatives in so that the flat part of is the unit line segment on the axis, is contained in the upper halfplane, and the highest coordinate of any point in is attained at . This is illustrated in Figure 6.
By Lemma 3.3, we conclude that implies that every horizontal chord in has length . This implies that for any coordinate of a point in . On the other hand, contains a triangle of base and height , so by comparing areas we must have . Thus we conclude that . It is well known that the space of convex sets of fixed volume in some bounded region of given the Hausdorff topology is compact (this is the socalled Blaschke selection theorem). Thus we conclude that there exists a subsequence of that converges. {definition} We define the complex configuration space for and as follows. We let denote a particular subset of . An ordered pair is in if and only if it satisfies the following:

.

If is the halfplane of , then .

If is the halfplane of , then .

It holds that .
We then give the subspace topology.
One can see that the middle two conditions taken together just mean that and coincide on , and the last condition just says that precedes if we traverse their convex boundary counterclockwise. Examples appear in Figure 5(b) and in Figure 8.
Lemma 5.2
The space is compact.
Let be a sequence of elements of the quotient . Lift these to a sequence in where we assume (after passing to a subsequence using Lemma 5.1) that is convergent to .
Now refer to Figure 7. Label the intersection of the flat boundary of with the convex boundary of as . Label the intersection of the flat boundaries of and as . Label the intersection of the flat boundary of with its convex boundary other than as . Clearly we can extract a subsequence for which converges to a point on the convex boundary of , and then extract a further subsequence for which converges to a point on the flat boundary of . The only subtlety in this proof is to observe that shows that and are not on the corners of .
Given and , the boundedness of the area of implies that is bounded, so we extract another subsequence for which additionally converges to a point . Now it is easy to see that the fixing of provide only a bounded set for to range over, so compactness follows again using the Blaschke selection theorem.
Lemma 5.3
There exists an absolute constant such that if we are given and angles with , then we can find a sequence so that and .
Let . Now define inductively for as follows. The function
(19) 
is strictly decreasing until it reaches zero, where it remains constant. Thus there exists a unique so that . We now have an infinite chain of angles so that has area for . This is illustrated in Figure 5.
Let be the maximum index such that . Note that since is compact, there exist absolute constants (not depending on ) such that
(20) 
for all . Thus we conclude that
(21) 
which is sufficient.
6 A moment estimate
An ingredient in the central limit theorems for the polygonal case is a moment estimate groen1 (), page 341, Lemma 2.5, and groen2 (), page 36, Lemma 2.1. Here, we prove an analogous estimate in general.
Proposition 6.1
Let denote either or . There exist absolute constants and such that for any convex and interval with , we have the following estimate:
(22) 
We can split up into subintervals of small affine invariant measure, and use Cauchy’s inequality,
(23) 
so it suffices to show that there exist and so that for all and satisfying , it holds that the moment generating function is for all .
Since is compact, the affine invariant measure of the interval between the angles of and is bounded below. Thus we conclude that it suffices to show that for every , the moment generating function of is defined in a neighborhood of zero where is the angle of and is the angle of .
Now we may put such an element in a standard position in by requiring that both boundary segments have equal length, and that the angles of and are and , respectively, (see Figure 8).
Thus, given the configuration in Figure 8, we would like to show that for sufficiently small , we have . First, write
(24)  
If , then is bounded by the number of points of the Poisson process in the region . An elementary calculation shows that , where is a Poisson distribution of parameter . We may assume , so . Thus in this case
(25)  
If , then is bounded by , so we have
(26)  
By compactness of , the angle where the convex part of meets the flat boundary of is bounded below by an absolute constant (say by , see Figure 8). Similarly, the lengths of the flat parts of and are bounded above absolutely (say by ). Thus the area above the dotted line in Figure 8 is bounded above absolutely, say by .
Now we claim that
(28) 
[recall that is the length of where is the horizontal line passing through ]. If , then the area of is by definition. If , then argue as follows: the area of above the dotted line is certainly less than , and the area of below the dotted line is bounded by .
7 A dependence estimate
{definition}If is an interval, then we let be the algebra which keeps track of for .
For example, is measurable if and only if .
The type of dependence estimate we prove will be an mixing estimate, that is, an estimate on where and are events that are supposed to be almost independent. This type of estimate has been used previously in studying random polygons; we were motivated to prove our estimate by a similar result in groen1 (), page 341, Theorem 2.3.
Lemma 7.1
Let and be two disjoint intervals in . Let and . Then
(32)  
The proof is an elementary calculation and is given in the Appendix. The object of this section is to reexpress the righthand side of (7.1) in terms of the affine invariant measure.
Lemma 7.2
There exists an absolute constant such that if , then area of is .
We use Lemma 5.3 to construct a sequence so that and . From this decomposition, we see that it suffices to show that there exists such that for all
(33) 
Now we know that for some halfplane and that additionally . Hence it suffices to show that
(34) 
whenever and .
Remember that and have intersection . Thus it suffices to show that for every , the following is true:
(35) 
whenever and [see Figure 9(a)]. Here, if we put and in standard position (i.e., as in Figure 9, with both flat boundaries of equal length), then has negative slope. Denote by the intersection of the flat boundaries of and . Then since , we must have . From this, we see that , so we may rewrite (35) as
(36) 
The minimum of this expression is clearly a continuous function on , and is by definition invariant under the action of . We know that is compact, so it suffices to show that for any fixed configuration , expression (36) is bounded below away from zero. Certainly, if this ratio were approaching zero, then . However in this case, the situation is illustrated in Figure 9(b), where it is clear that ratio (36) in fact does not approach zero, but rather some appropriate ratio of lengths of the boundaries of the caps. Thus we are done.
Lemma 7.3
There exists an absolute constant such that if
(37) 
We pick the unique so that and , .
Define
(38) 
so .
Now if for all , then by Lemma 4.2, the area of is . The same applies if for . Thus in both of these cases, we conclude that or .
If for some and for some , then necessarily . Thus we know that for all , at least one of the following is true:

,

,

.
By elementary integration, the integral over the second and third regions is . The area of the first region is by Lemma 7.2, so we are done.
Proposition 7.4
There exists an absolute constant so that if and are two disjoint intervals in , and we have events and , then
(39) 
where denotes if and if instead .
The reader may wonder exactly what follows from an mixing estimate. We won’t answer that here, though we will record here two lemmas that will be useful later whose hypotheses are mixing estimates.
Lemma 7.5 ((centrallimittheorembook (), page 115, Lemma 1(6)))
Suppose and are random variables taking values in such that
(40) 
for all . Then we have
(41) 
Lemma 7.6
Suppose and are random variables taking values in such that
(42) 
for all . Let , and let equal the sum of independent copies of and . Then we have
(43) 
Let be any finite increasing sequence of real numbers. Then we have
Now using the definition of , we can bound this below by
Thus we find that
(46)  
Now choose real numbers so that the probability that falls in the open interval