From Steiner formulas for cones to concentration of intrinsic volumes

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4 M. B. MCCOY AND J. A. TROPP3.1. The master Steiner formula. Let us introduce a class of Gaussian integrals. Fix a Borel measurablebivariate function f : R 2 + → R. Consider the geometric functionalϕ f : C d → R where ϕ f (C) := E [ f ( ‖Π C (g )‖ 2 , ‖Π C ◦(g )‖ 2 )] . (3.1)Throughout the paper, we reserve the letter g for a standard normal random vector whose dimension isdetermined by context, and all expectations are interpreted as Lebesgue integrals. We can develop an elegantexpansion of ϕ f in terms of the conic intrinsic volumes.Theorem 3.1 (Master Steiner formula for cones). Let f : R 2 + → R be a Borel measurable function. Then thegeometric functional ϕ f defined in (3.1) admits the expressionϕ f (C) =d∑ϕ f (L k ) · v k (C) for C ∈ C d (3.2)k=0provided that all the expectations in (3.2) are finite. Here, L k denotes a k-dimensional subspace of R d and theconic intrinsic volumes v k are introduced in Definition 2.2.The coefficients ϕ f (L k ) in the expression (3.2) have an alternative form that is convenient for computations.Let L k be an arbitrary k-dimensional subspace of R d . According to the definition (3.1) of the functional ϕ f ,ϕ f (L k ) = E [ f ( ‖Π Lk (g )‖ 2 , ‖Π Lk◦(g )‖ 2 )] .The marginal property of the standard normal vector g ensures that Π Lk (g ) and Π Lk◦(g ) are independentstandard normal vectors supported on L k and L k ◦ . Thus, ‖Π Lk (g )‖ 2 and ‖Π Lk◦(g )‖ 2 are independent chi-squarerandom variables with k and d − k degrees of freedom respectively. Note the convention that a chi-squarevariable with zero degrees of freedom is identically zero. In view of this fact, we have the following equivalentof Theorem 3.1.Corollary 3.2. Instate the hypotheses and notation of Theorem 3.1. Let { X 0 ,..., X d} be an independent sequenceof random variables where X k has the chi-square distribution with k degrees of freedom, and let { X ′ 0 ,..., X ′ d} bean independent copy of this sequence. Thenϕ f (C) =d∑E [ f ( X k, Xd−k)] ′ · vk (C) for C ∈ C d .k=0Corollary 3.2 has an appealing probabilistic consequence. For every cone C ∈ C d , the random variable‖Π C (g )‖ 2 is a mixture of chi-square random variables X 0 ,..., X d where the mixture coefficients v k (C) aredetermined solely by the cone.We outline the proof of Theorem 3.1 in Sections 7 and 8. The argument involves techniques that arealready familiar to experts. Many of the core ideas appear in McMullen’s influential paper [McM75]. Asimilar approach has been used in hyperbolic integral geometry [San80, p. 242]; see also the proof of [SW08,Thm. 6.5.1]. The only technical novelty is our method for showing that the conic intrinsic volumes arecontinuous with respect to the conic Hausdorff metric.3.2. How big is the expansion of a cone? The Euclidean Steiner formula (1.1) describes the volume of aEuclidean expansion of a compact convex set. Although it may not be obvious from the identity (3.2), themaster Steiner formula contains information about the volume of an expansion of a convex cone. This sectionexplains the connection, which justifies our decision to call Theorem 3.1 a Steiner formula.First, we argue that there is a simple expression for the Gaussian measure of a Euclidean expansion of aconvex cone.Proposition 3.3 (Gaussian Steiner formula). For each cone C ∈ C d and each number λ ≥ 0,γ d (C + λB d ) = P { dist 2 (g ,C) ≤ λ } =d∑P { X d−k ≤ λ } · v k (C) (3.3)where γ d is the standard normal measure on R d and B d is the unit ball in R d . The random variable X k followsthe chi-square distribution with k degrees of freedom.k=0
STEINER FORMULAS FOR CONES AND INTRINSIC VOLUMES 5Proof. The first identity in (3.3) is immediate. For the second, we appeal to Corollary 3.2 with the function{1, b ≤ λf (a,b) =0, otherwise.This step yields the relationP { dist 2 (g ,C) ≤ λ } = P { ‖Π C ◦(g )‖ 2 ≤ λ } =d∑E [ f ( X k , X ′ )] d∑d−k = P { X ′ d−k ≤ λ} · v k (C).k=0The first equality depends on the representation (7.3) of the distance to a cone in terms of the metric projectoronto the polar cone. The result (3.3) follows because X ′ d−k has the same distribution as X d−k.□We can also establish the spherical Steiner formula (1.2) as a consequence of Theorem 3.1 by replacing theGaussian vector g in Proposition 3.3 with a random vector θ that is uniformly distributed on the Euclideanunit sphere. This strategy leads to an expression for the proportion of the sphere subtended by an angularexpansion of the cone.Proposition 3.4 (Spherical Steiner formula). For each cone C ∈ C d and each number λ ∈ [0,1], it holds thatP { dist 2 (θ,C) ≤ λ } =k=0d∑P { B d−k,d ≤ λ } · v k (C). (3.4)k=0The random vector θ is drawn from the uniform distribution on the unit sphere S d−1 in R d , and the randomvariable B j,d follows the ( 1BETA2 j, 12 d) distribution.Proof. When λ = 1, both sides of (3.4) equal one. For λ < 1, we convert the spherical variable to a Gaussianusing the relation θ ∼ g /‖g ‖ where ∼ denotes equality of distribution. It follows thatP { dist 2 (θ,C) ≤ λ } = P { ‖Π C ◦(g )‖ 2 ≤ λ(1 − λ) −1 · ‖Π C (g )‖ 2 } .This identity depends on the representation (7.3) of the distance, the nonnegative homogeneity of the metricprojector Π C ◦, and the Pythagorean relation (7.4). Apply Corollary 3.2 with the function{1, b ≤ λ(1 − λ) −1 af (a,b) =0, otherwise.To finish, we recall the geometric interpretation [Art02] of the beta random variable: B j,d ∼ ‖Π L j(θ)‖ 2 whereL j is a j-dimensional subspace of R d .□4. PROBABILISTIC ANALYSIS OF INTRINSIC VOLUMESIn this section, we sic probabilistic methods on the conic intrinsic volumes. Our main tool is the masterSteiner formula, Theorem 3.1, which we use repeatedly to convert statements about the intrinsic volumes of acone into statements about the projection of a Gaussian vector onto the cone. We may then apply methodsfrom Gaussian analysis to study this random variable.4.1. The intrinsic volume random variable. Definitions 2.1 and 2.2 make it clear that the intrinsic volumesform a probability distribution. This observation suggests that it would be fruitful to analyze the intrinsicvolumes using techniques from probability. We begin with the key definition.Definition 4.1 (Intrinsic volume random variable). Let C ∈ C d be a closed convex cone. The intrinsic volumerandom variable V C has the distributionP { V C = k } = v k (C)for k = 0,1,2,...,d.Notice that the intrinsic volume random variable of a cone C ∈ C d and its polar have a tight relationship:P { V C ◦ = k } = v k (C ◦ ) = v d−k (C)because polarity reverses the sequence of intrinsic volumes. In other words, V C ◦ ∼ d −V C .
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