Threshold Phenomena for Random Cones

We consider an even probability distribution on the d-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given N independent random vectors with this distribution, under the condition that they do not positively span the whole space, the positive hull of these vectors is a random polyhedral cone (and its intersection with the unit sphere is a random spherical polytope). It was first studied by Cover and Efron. We consider the expected face numbers of these random cones and describe a threshold phenomenon when the dimension d and the number N of random vectors tend to infinity. In a similar way we treat the solid angle, and more generally the Grassmann angles. We further consider the expected numbers of k-faces and of Grassmann angles of index d-k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d-k$$\end{document} when also k tends to infinity.

amount to a rapid shift in the likelihood of a property's occurrence when a dimension parameter crosses a critical level (a threshold)." Two early observations in highdimensional random geometry of such distinctly different behavior below and above a threshold were published in 1992. Dyer et al. [10] considered the convex hull of N = N (d) points chosen independently at random (with equal chances) from the vertices of the unit cube in R d . Let V d,N denote the volume of this random polytope. Then, for every ε > 0, Here E denotes mathematical expectation. The paper [10] has a similar result for the convex hull of i.i.d. uniform random points from the interior of the unit cube. We have quoted this example as an illustration of what we have in mind: for instance, a d-dimensional random polytope with its number N of vertices depending on d, where a small change of this dependence causes an abrupt change of some property as d → ∞. In the work of Vershik and Sporyshev [20], a d-dimensional random polytope is obtained as a uniform random orthogonal projection of a fixed regular simplex with N vertices in a higher-dimensional space, and threshold phenomena are exhibited for the expected numbers of k-faces, under the assumption of a linearly coordinated growth of the parameters d, N , k. Similar models, also with the regular simplex replaced by the regular cross-polytope, and random projections extended to more general random linear mappings, have found important applications in the work of Donoho and collaborators. We refer to the article of Donoho and Tanner [8], where also earlier work of these authors is cited and explained. The paper [9] of the same authors treats random projections of the cube and the positive orthant in a similar way. Generally in stochastic geometry, threshold phenomena have been investigated for face numbers, neighborliness properties, volumes, intrinsic volumes, more general measures, and for several different models of random polytopes. Different phase transitions were exhibited. We mention that [11] has extended the model of [10] by introducing more general distributions for the random points. The paper [17] considers convex hulls of i.i.d. random points with either Gaussian distribution or uniform distribution on the unit sphere. In [3,4], the points have a beta or beta-prime distribution. The paper [5] studies facet numbers of convex hulls of random points on the unit sphere in different regimes. The papers [3,13,17] deal also with polytopes generated by intersections of random closed halfspaces. An important role is played by phase transitions in convex programs with random data. We quote from the work of Amelunxen et al. [1], which discovers and describes several of these phenomena: "This paper provides the first rigorous analysis that explains why phase transitions are ubiquitous in random convex optimization problems . . .Ṫhe applied results depend on foundational research in conic geometry." In this paper, we consider a model of random polyhedral convex cones (or, equivalently, of random spherical polytopes) that was introduced by Cover and Efron [6] (and more closely investigated in [14]). Let φ be a probability measure on the Euclidean space R d that is even (invariant under reflection in the origin o) and assigns measure zero to each hyperplane through the origin. For n ∈ N, the (φ, n)-Cover-Efron cone C n is defined as the positive hull of n independent random vectors X 1 , . . . , X n with distribution φ, under the condition that this positive hull is different from R d . The intersection C n ∩ S d−1 with the unit sphere S d−1 is a spherical random polytope, contained in some closed hemisphere. In the following it will be convenient to work with polyhedral cones instead of spherical polytopes.
For k ∈ {1, . . . , d − 1}, let f k (C n ) denote the number of k-dimensional faces of the cone C n (equivalently, the number of (k − 1)-dimensional faces of the spherical polytope C n ∩ S d−1 ). We are interested in the asymptotic behavior of the expectation E f k (C n ), as d tends to infinity and n grows suitably with d.

Convention
In the following, C N is a (φ, N )-Cover-Efron cone in R d , and N = N (d) ≥ d is an integer depending on the dimension d, but we will omit the dimension d in the notation.
Any k-face of the cone C N is a.s. the positive hull of k vectors from X 1 , . . . , X N , and there are N k possible choices. Therefore, we consider the quotient E f k (C N )/ N k . For this, we claim the following behavior above and below a threshold.
with a number δ ∈ [0, 1]. Let k ∈ N be fixed. Then the Cover-Efron cone C N satisfies We have assumed here that the number k is constant. But if in the case where 0 ≤ δ < 1/2 we let also k depend on d, in such a way that (thus, k grows with d, but sublinearly) then This is easily read off from the proof of the second part of Theorem 1.1 given in Sect. 4.
The case 1/2 < δ ≤ 1 of Theorem 1.1 follows from the stronger Theorem 1.2 below, but we have formulated Theorem 1.1 in this way to illuminate the different behavior if δ is below 1/2 or above 1/2. Partial information on the case δ = 1/2 is contained in Theorem 1.2. In this theorem, also the number k is allowed to depend on d. Then with constants a > 0 and α ≥ 0.
The assumptions of this theorem are, in particular, satisfied if k is constant and On the other hand, Theorem 1.7 below implies that if As another functional of a closed convex cone C ⊂ R d , we consider the solid angle v d (C). This is the normalized spherical Lebesgue measure of C ∩ S d−1 . (We avoid the notation V d (C) used in [14], since V d is often used for the volume of a convex body. The reader is warned that what we denote here by v d (C) was denoted by v d−1 (C ∩ S d−1 ) in [18,Sect. 6.5].) More generally, we consider the Grassmann angles. For a closed convex cone C ⊂ R d which is not a subspace, the jth Grassmann angle of C, for j ∈ {1, . . . , d −1}, is defined by The latter is the unique Haar probability measure on G(d, d − j), the Grassmannian of (d − j)dimensional linear subspaces of R d . Thus, Grassmann angles were introduced by Grünbaum [ We have assumed here that the number k is constant. But if in the case where 0 ≤ δ < 1/2 we let also k depend on d, in such a way that (thus, k grows with d, but sublinearly) then This can be seen from the second part of the proof of Theorem 1.4 given in Sect. 5.
Again, the first part of this theorem has a stronger version, given by the following theorem.
with some real constant a. Then with constants a > 0 and α ≥ 0.
And similarly as above, there is a more precise asymptotic relation if N = 2d.
On the other hand, Theorem 1.9 below implies that if Clearly, in Theorem 1.1 (and similarly in Theorem 1.4), the change when passing a threshold is not so abrupt as in the examples from [9], where both parameters, N and k, grow linearly with the dimension: below the threshold δ = 1/2, the limit in question increases (decreases) with the parameter to an extremal value, above the threshold, it remains constant. The situation changes if also the number k increases sublinearly in the dimension. Then indeed we also have a sharp threshold as pointed out above. Now we consider the case where k increases proportionally to the dimension; then a more subtle phase transition is observed. Under a linearly coordinated growth, for k-faces we find the same threshold as established by Donoho and Tanner [9] in their investigation of random linear images of orthants. This may seem unexpected, since the random cones considered in [9] and here have different distributions (see, however, the appendix).
As in [9], we define (the index W stands for 'weak' threshold). Then We note that the first assumption of this theorem, ρ < ρ W (δ), implies that for large d we have N + k < 2d.
Adapting an argument of Donoho and Tanner [9] to the present situation, we can also replace the convergence of an expectation in the first part of Theorem 1.7 by the convergence of a probability, at the cost of a smaller threshold.
Then there exists a positive number ρ S (δ) such that, for ρ < ρ S (δ), There is also a counterpart to Theorem 1.7 for Grassmann angles. Then After some preliminaries in the next section, we collect a number of auxiliary results about sums of binomial coefficients in Sect. 3. Then we prove the first three theorems in Sect. 4, Theorems 1.4 to 1.6 in Sect. 5, Theorems 1.7 and 1.8 in Sect. 6, and Theorem 1.9 in Sect. 7.

Preliminaries
First we recall two classical facts. For n ∈ N, let H 1 , . . . , H n ∈ G(d, d − 1). Assume that these hyperplanes are in general position, that is, the intersection of any m ≤ d of them is of dimension d −m. Then the number of d-dimensional cones in the tessellation of R d induced by these hyperplanes is given by From this result of Steiner (in dimension three) and Schläfli, Wendel has deduced the following. If X 1 , . . . , X n are i.i.d. random vectors in R d with distribution φ (enjoying the properties mentioned above), then where P stands for probability and pos denotes the positive hull. For references and proofs, we refer to [18, Sect. 8.2.1]. Now we can write down the distribution of the (φ, n)-Cover-Efron cone C n , namely where C d denotes the space of closed convex cones in R d (with the topology of closed convergence) and B(C d ) is its Borel σ -algebra. There is an equivalent representation of C n . For this, we denote by φ * the image measure of φ under the mapping x → x ⊥ from R d \{o} to G(d, d −1). Let H 1 , . . . , H n be i.i.d. random hyperplanes with distribution φ * . They are almost surely in general position. The (φ * , n)-Schläfli cone S n is obtained by picking at random (with equal chances) one of the d-dimensional cones from the tessellation induced by H 1 , . . . , H n . Its distribution is given by where S • n denotes the polar cone of S n .

Auxiliary Results on Binomial Coefficients
First we collect some information on the Wendel probabilities Let ξ n be a random variable having the binomial distribution with parameters n and 1/2, i.e., Thus we can write and therefore, by (1), Similarly, by (2) we have The following two lemmas concern the Wendel probabilities and are therefore stated here, although they are not needed before the proof of Theorem 1.8.
This and induction can be used to prove that for k ∈ {1, . . . , d − 1}, For j ∈ {1, . . . , k} we have This gives and thus the assertion.
Hence, for arbitrary p ≥ 1 we may write Below some information on binomial coefficients is required. First we note Stirling's formula n! = √ 2π n e −n n n e θ/12n , 0 < θ < 1.
It implies, in particular, that (where a n ∼ b n as n → ∞ means that lim n→∞ a n /b n = 1).
Proof Assume that d/N → δ as d → ∞, with a number 0 ≤ δ < 1/2. We write N = αd, where α depends on d and satisfies α → δ −1 as d → ∞. If δ = 0, this means that α → ∞. We assume that d is so large that α > 2. From (14) we have We conclude that lim sup Together with (15) this completes the proof.
We state another simple lemma.
which gives the first relation. If y : which gives the second relation.

Proofs of Theorems 1.1 to 1.3
Proof of Theorem 1.1 As already mentioned, the first part of Theorem 1.1 follows from Theorem 1.2, which will be proven below. To prove the second part of Theorem 1.1, we assume that d/N → δ as d → ∞, where 0 ≤ δ < 1/2. We write (1) in the form and here Since also (d − k)/(N − k) → δ, we deduce from Lemma 3.4 that the normalized sums in the numerator and denominator of (16) tend to the same finite limit. It follows that lim d→∞ E f k (C N )/ N k = (2δ) k . Proof of Theorem 1.2 As in Sect. 3, we denote by ξ n a random variable having the binomial distribution with parameters n ∈ N and p = 1/2. Let ξ * n denote the standardized version of ξ n , that is, ξ * n = (ξ n − Eξ n )/ √ Vξ n with Eξ n = n/2 and Vξ n = n/4. Then (5) implies that by the Berry-Esseen Theorem (see, e.g., Shiryaev [19, p. 426]), where is the distribution function of the standard normal distribution. We have we get for (k + 1)/N ≤ 1/2 (which holds if d is large enough) that with θ ∈ [1,4]. This shows that We define In the same way we get where θ ∈ [1, 4]. Thus we arrive at In view of (17), it is sufficient to show that To verify this, we use the fact that (the difference of the function on the left-hand side and the right-hand side converges to zero, as x → ∞, the derivative of this difference is non-positive for x ≥ 1). Applying this inequality with x = −a(d), we get Since a(d) 2 as d → ∞, the assertion follows. (1) we have

Proof of Theorem 1.3 Suppose that N = 2d. Then by
We distinguish two cases. If k is odd, say k = 2 − 1 with ∈ N, then where Lemma 3.5 was used. Since Stirling's formula (8) yields it follows (again using Lemma 3.5) that If k is even, say k = 2 with ∈ N, then by Lemma 3.5. By Stirling's approximation (8), and hence we get Thus in both cases the asymptotic relation is proven.

Proofs of Theorems 1.4 to 1.6
Proof of Theorem 1.4 The first part of Theorem 1.4 follows from Theorem 1.5, which will be proven below. For the second part of the proof, we assume that d/N → δ as d → ∞, with 0 ≤ δ < 1/2. We note that relation (2) shows that Then and here The assertion follows.
Proof of Theorem 1. 5 The random variables ξ n and ξ * n are defined as in the proof of Theorem 1.2. Using (6), we proceed as in that proof and obtain The rest of the proof follows that for Theorem 1.2.

Linearly Growing Face Dimensions
In this section and the next one, we allow also the number k to grow linearly with the dimension d. In the present section, we are interested in a phase transition for the expectation E f k (C N )/ N k . It turns out that it appears at the same threshold as it was observed earlier by Donoho and Tanner [9] for a different, but closely related class of random polyhedral cones. These authors considered a real random d × N matrix A of rank d, where d < N , the nonnegative orthant of R N , and its image AR N + in R d . Considering the column vectors of A as random vectors in R d , the image AR N + is the positive hull of these vectors. For a suitable distribution, the random cone AR N + is obtained in a similar way as the Cover-Efron cone, just by omitting the condition that the cone is different from R d . Imposing this condition leads, of course, to different distributions of the random cones. Comparing formula (13) of [9] with our formula (1), where the right-hand side can be written as we see that it results in an additional denominator in the expression for the expected number of k-faces, thus increasing this expectation. Therefore some of our estimates, though leading to the same threshold, require more effort.
Proof of Theorem 1.7 First we assume that 0 ≤ ρ < ρ W (δ). For this part of the proof, we reproduce an argument which was suggested by an anonymous referee (our original proof can be found at arXiv:2004.11473v1). We use the representation given by (5), where ξ n is a random variable with binomial distribution with parameters n and 1/2. Since ξ n is in distribution equal to the sum of n i.i.d. Bernoulli random variables with parameter 1/2, the weak law of large numbers gives lim n→∞ P ξ n > n 1 2 Therefore, as stated.
For the second part of the proof, we use (16) and show first that for increasing d the terms in brackets remain between two positive constants. The asymptotic behavior of the remaining quotient is then determined with the aid of Stirling's formula.
We assume that ρ > ρ W (δ). Then ρ > 2 − δ −1 , irrespective of whether δ ≥ 1/2 or not. For sufficiently large d (which we assume in the following), we then have N − 2d + k > 0. Thus, we can apply (14) to the normalized sum in the numerator of (16). This yields Here, where the last denominator is positive. It follows that for all sufficiently large d. Here and below we denote by c i a positive constant that is independent of d.
In view of (16), we now determine the asymptotic behavior of Here, To treat the remaining terms, we use the Stirling formula (8). Define where ϕ is contained in a fixed interval independent of d, and We define Note that for a, b H (a, b). Differentiation yields If a ≤ 1/2, then 1). On the other hand, if a > 1/2 and b > 2 − a −1 , we have Since the function g defined by (21) satisfies g(1/2) = 1 and g (a) > 0 for 1/2 < a < 1, we also have Now we distinguish two cases.
(i) Let δ ≤ 1/2. Then (16), (19), and (20) yield To estimate the last binomial coefficient, we use Stirling's approximation (8) together with (22). Thus, we get for large d the lower bound Combining these estimates and starting again from (16), we finally obtain Here we have used that by (22) and that by (23). This completes the proof also in the case ρ > ρ W (δ).
Remark Under the assumption d/N → δ > 1/2, d → ∞, we have seen in the first part of the preceding proof that P(pos {X 1 , . . . , as a simple consequence of the weak law of large numbers. In this situation, an application of a large deviation (concentration) result for the binomial distribution in fact shows that the convergence is exponentially fast. For this, we choose d sufficiently large so that d/(N − 1) ≥ 1/2. Then Hence, if d/N → δ < 1/2 and d is large enough so that In this case, a finer analysis of the ratio (18) with corresponding lower bounds for the involved probabilities is required.

Proof of Theorem 1.8 Again we define
Let δ > 1/2 and 0 < ρ < ρ S (δ). Then G(δ, ρ) < 0. For sufficiently large d, we have G(δ d , ρ d ) < 0 as well as δ d > 1/2 and ρ d < ρ S (δ d ). We assume that d is large enough in this sense. Since ρ s (δ d ) < ρ W (δ d ), we have N − 2d + k < 0. Let Therefore, (7) (and with the notation used there). By Boole's inequality, and thus Here, By Lemmas 3.1 and 3.2, Since N − 2d + k − 1 < 0 for all d, the sum in the last numerator of (25) is zero, hence Using this and the identities together with (8), we get where (24) was used and where ϕ is in a fixed interval. Since G(δ d , ρ d ) → G(δ, ρ) < 0 as d → ∞, it follows that from which the assertions follow.
7 Proof of Theorem 1.9 We use the representations and show the convergence of the quotients, under different assumptions.
Finally, we observe that (1 − ρ) < 1/2 and g(δ(1 − ρ)) > 1. Thus, (29) is obtained again. Theorem 1.7. Then, N is between d and (2 − ε)d, which means that the probability that the positive hull of the vectors X 1 , . . . , X N is R d goes to 0 exponentially fast. So, both models of cones differ just on an event of exponentially small probability and are equal otherwise. Moreover, the events of exponentially small probability make no contribution to the expected f -vector because on this event the Donoho-Tanner cone is R d and f k = 0 for all k < d. The case δ < 1/2 is more difficult to explain." (end of quotation) The referee then sketches an argument for this case which, in his/her opinion, is not rigorous, but makes the result quite natural.