Facets of high-dimensional Gaussian polytopes

We study the number of facets of the convex hull of n independent standard Gaussian points in R d . In particular, we are interested in the expected number of facets when the dimension is allowed to grow with the sample size. We establish an explicit asymptotic formula that is valid whenever d/n → 0. We also obtain the asymptotic value when d is close to n .


Introduction
The convex hull [X 1 , . . ., X n ] of n independent standard Gaussian samples X 1 , . . ., X n from R d is the Gaussian polytope P as n tends to infinity are well understood by now.For i = 0 . . ., d and polytope Q, let f i (Q) denote the number of i-faces of Q and let V i (Q) denote the ith intrinsic volume of Q.The asymptotic behavior of the expected value of the number of facets n ) as n → ∞ was provided by Rényi, Sulanke [22] if d = 2, and by Raynaud [21] if d ≥ 3. Namely, they proved that, for any fixed d, as n → ∞.For i = 0, . . ., d, expected value of V i (P n ) as n → ∞ was computed by Affentranger [1], and that of f i (P (d) n ) was determined Affentranger, Schneider [2] and Baryshnikov, Vitale [3], see Hug, Munsonius, Reitzner [15] and Fleury [12] for a different approach.More recently, Kabluchko and Zaporozhets [18,19] proved explicit expressions for the expected value of V d (P ).Yet these formulas are complicated and it is not immediate how to deduce asymptotic results for large n high dimensions d.
After various partial results, including the variance estimates of Calka, Yukich [6] and Hug, Reitzner [16], central limit theorems were proved for f i (P Bárány and Vu [4], and for V i (P (d) n ) by Bárány and Thäle [5].These results have been strengthened considerably by Grote and Thäle [14].The interesting question whether Ef d−1 (P (d) n ) is an increasing function in n was answered in the positive by Kabluchko and Thäle [17].It would be interesting to investigate the monotonicity behavior of the facet number if n and d increases simultaneously.
The "high-dimensional" regime, that is, when d is allowed to grow with n, is of interest in numerous applications in statistics, signal processing, and information theory.The combinatorial structure of P (d) n , when d tends to infinity and n grows proportionally with d, was first investigated by Vershik and Sporyshev [23], and later Donoho and Tanner [11] provided a satisfactory description.For any t > 1, Donoho, Tanner [11] determined the optimal ̺(t) ∈ (0, 1) such that if n/d tends to t, then P ). See Donoho [10], Candés, Romberg, and Tao [7], Candés and Tao [8,9], Mendoza-Smith, Tanner, and Wechsung [20].
In this note, we consider f d−1 (P n ), the number of facets, when both d and n tend to infinity.Our main result is the following estimate for the expected number of facets of the Gaussian polytope.The implied constant in O(•) is always some absolute constant.We write lln x for ln(ln x).
When n/d tends to infinity as d → ∞, Theorem 1.1 provides the asymptotic formula .
If n/(de d ) → ∞, then we have d ln n d → 0 and hence as d → ∞.In the case when n grows even faster such that (ln n)/(d ln d) → ∞, the asymptotic formula simplifies to the result (1) of Rényi, Sulanke [22] and Raynaud [21] for fixed dimension.
Corollary 1.2.Assume P There is a (simpler) counterpart of our main results stating the asymptotic behavior of the expected number of facets of P as d → ∞.
This complements a result of Affentranger and Schneider [2] stating the number of k-dimensional faces for k ≤ n − d and n − d fixed, In the next section we sketch the basic idea of our approach, leaving the technical details to later sections.In Section 3 we provide asymptotic approximations for the tail of the normal distribution.In Section 4 concentration inequalities are derived for the β-distribution.Finally, in Sections 5 and 6, Corollary 1.2 and Theorem 1.3 are proven.

Outline of the argument
Our proof is based on the approach of Hug, Munsonius, and Reitzner [15].
In particular, [15,Theorem 3.2] states that if n ≥ d + 1 and X 1 , . . ., X n are independent standard Gaussian points in R d , then Note that similar integrals appear in the analysis of the expected number of k-faces for values of k in the entire range k = 0, . . ., d − 1.In our case, the analysis boils down to understanding the integral of Φ(y) n−d φ(y) d over the real line.By substituting (1 dominating, and we need to investigate the asymptotic behavior of φ(Φ −1 (1− u)) as u → 0. We show that the essential term is precisely 2u.Hence, it makes sense to rewrite the integral as du .
For x, y > 0, the Beta-function is given by B(x, y) With this, we have established the following identity: Proposition 2.1. where In Lemma 3.3 below we show that

Asymptotics of the Φ-function
To estimate Φ(z), we need a version of Gordon's inequality [13] for the Mill's ratio: Lemma 3.1.For any z > 1 there exists θ ∈ (0, 1), such that Proof.It follows by partial integration that which yields the lemma.
Lemma 3.2.For any u ∈ (0, e −1 ] there is a δ = δ(u) ∈ (0, 16) such that (5) Proof.It is useful to prove (5) for the transformed variable u = e −t .We define which exists for t > 0. In a first step we prove that this is the asymptotic expansion of . In a second step we show the bound on δ.Observe that z ≥ 1 implies t ≥ ln Φ(−1)) = −2, 54 . . . .By Lemma 3.1, for z ≥ 1 as z → ∞ with some θ(z) ∈ (0, 1), which immediately implies that z = z(t) → ∞ as t → ∞.Equation (7) shows that e t ≥ 2 √ πze z 2 and thus for z ≥ 1.The function z = z(t) is the inverse function we are looking for, if it satisfies We plug (6) into this equation.This leads to and shows − ln(2 . Thus the function z(t) given by ( 6) in fact satisfies (7) and therefore it is the asymptotic expansion of the inverse function.

Concentration of the β-distribution
A basic integral for us is the Beta-integral Let U ∼ B(α, β) distributed.Then EU = β α+β and var(U) = Next we establish concentration inequalities for a Beta-distributed random variable around its mean.Observe that if U ∼ B(α, β), then 1−U ∼ B(β, α).Hence we may concentrate on the case α ≥ β.
In the last step we use Stirling's formula, to see that Lemma 4.2.Let U ∼ B(a+1, b+1) distributed with a ≥ b and set n = a+b.
Then for λ ≥ 2, Proof.We assume that a ≥ b and thus a ≥ n 2 .We have to estimate the probability The use of the binomial formula and the Gamma functions yields Using (11) this gives

The case n − d large
In this section we combine Lemma 3.3 which gives the asymptotic behavior of g d (u) as u → 0, with the concentration properties of the Beta function just obtained.We split our proof in two Lemmata.These two bounds prove Theorem 1.1.The idea is to split the expectation into the main term close to d n and two error terms, for n ≥ 10d.The probability that U is small is estimated by Lemma 4.1 with for d ≥ 6. Combining both estimates and using for x ∈ [0, 1  2 ], we have for for n ≥ e e d.Here, note that lln x ln x is decreasing for x ≥ e e .Now using since δ ≥ 0, and where the last term follows from (12).For the first term we use that φ(Φ −1 (•)) is a symmetric and concave function and thus increasing on [0, e −2 d n ], and that δ ≥ 0.
Now the remaining integration is trivial.We use Stirling's formula (11) to estimate the Beta-function and obtain .
This yields 6 The case n − d small Finally, it remains to prove Theorem 1.3.The starting point here is again formula (2), together with the substitution y → y √ d .
We need again estimates for the logarithm, namely ln(1 + x) = x − θ 3 x 2 < x with some θ 3 = θ 3 (x) ≥ 0. In addition, there exists c 3 ∈ R such that θ fixed dimension d, the face numbers and intrinsic volumes of P (d) n (d) n ) and the number of k-faces f k (P (d) n

Theorem 1 . 1 .
Assume P (d) n is a Gaussian polytope.Then for d ≥ 78 and n ≥ e e d, we have

Theorem 1 . 3 .
Assume P (d) n is a Gaussian polytope.Then for n − d = o(d), we have Because the Beta function is concentrated around d n , see Lemma 4.1 and Lemma 4.2, this yields Eg d (U) ≈ ln n d which implies our main result.

Lemma 4 . 1 .( 1
Let U ∼ B(a+1, b+1) distributed with a ≥ b and set n = a+b.Then − x) a x b dx For an estimate from above we substitute x = b n for x ∈ (−1, 1].Since a ≥ b,

Lemma 5 . 1 .Lemma 5 . 2 .
For d ≥ d 0 = 78 and n ≥ e e d we have Eg d (U) ≤ e For d ≥ d 0 = 78 and n ≥ e e d we have Eg d (U) ≥ e