From Steiner formulas for cones to concentration of intrinsic volumes

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18 M. B. MCCOY AND J. A. TROPPDefine the conic Hausdorff metric dist H (C 1 ,C 2 ) between two cones C 1 ,C 2 ∈ C d bydist H (C 1 ,C 2 ) := inf { α ≥ 0 : T s (C 1 ,α) ⊃ C 2 and T s (C 2 ,α) ⊃ C 1}for C 1 ,C 2 ∈ C d .We equip C d with the conic Hausdorff metric and the associated metric topology to form a compact metricspace. It is not hard to check [Ame11, Prop. 3.2.4] that polarity is a local isometry on C d :For α < π/2, dist H (C 1 ,C 2 ) = α implies dist(C1 ◦ ,C 2 ◦ ) = α. (7.9)When we write expressions like C i → C for closed convex cones, we are always referring to convergence in theconic Hausdorff metric. The property (7.9) ensures that C i → C if and only if C ◦ i → C ◦ .A basic principle in the analysis of metric spaces is to identify a dense subset that consists of points withadditional regularity. We can make arguments that exploit this regularity and apply a limiting procedure toextend the claim to the rest of the space. To that end, let us demonstrate that the polyhedral cones form adense subset of C d . The approach mirrors [Sch93, Thm. 1.8.13].Fact 7.4 (Polyhedral cones are dense). Let C ∈ C d be a closed convex cone. For each ε > 0, there is a polyhedralcone C ε ∈ C d that satisfies dist H (C,C ε ) < ε.Proof sketch. We may assume C ≠ {0}. Let X be a finite ε-cover of the set C ∩S d−1 with respect to the angulardistance. That is,X = { x i : i = 1,..., N ε}⊂ C ∩Sd−1Consider the convex cone C ε generated by X :C ε := cone(X ) :=The cone C ε is polyhedral, and it satisfies dist H (C,C ε ) < ε.and min i dist s (x, x i ) < ε for all x ∈ C ∩S d−1 .{∑ }Nεi=1 τ i x i : τ i ≥ 0 .In order to perform limiting arguments, it helps to work with continuous functions. The next result ensuresthat projection onto a cone is continuous with respect to the conic Hausdorff metric.Fact 7.5 (Continuity of the projection). Consider a sequence (C i ) i∈N of closed convex cones in C d where C i → Cin the conic Hausdorff metric. For each x ∈ R d , the projection Π Ci (x) → Π C (x) as i → ∞.The proof is a straightforward exercise in elementary analysis, so we refer the reader to [McC13, Prop. 3.8]for details. This result has a Euclidean analog [Sch93, Lem. 1.8.9].8. PROOF OF THE MASTER STEINER FORMULAThis section contains the proof of Theorem 3.1. In Section 8.1, we establish a restricted version of themaster Steiner formula for polyhedral cones. In Section 8.2, we apply this basic result to prove that theintrinsic volumes of a closed convex cone are well defined, and we verify that the intrinsic volumes arecontinuous with respect to the conic Hausdorff metric. Afterward, we use an approximation argument toextend the master Steiner formula to closed convex cones in Section 8.3, and we remove the restrictions onthe function f in Section 8.4.8.1. Polyhedral cones and bounded continuous functions. We begin with a specialized version of themaster Steiner formula that restricts the cone C to be polyhedral and the function f to be bounded andcontinuous. This argument contains all the essential geometric ideas.Lemma 8.1 (Master Steiner formula for polyhedral cones). Let f : R 2 + → R be a bounded continuous function,and let C ∈ C d be a polyhedral cone. Then the geometric functional ϕ f defined in (3.1) admits the expressionϕ f (C) =d∑ϕ f (L k ) · v k (C) (8.1)k=0where L k is a k-dimensional subspace of R d and the conic intrinsic volumes v k are introduced in Definition 2.1.Proof. Define the random variables u = Π C (g ) and w = Π C ◦(g ). The tiling (7.7) induced by a polyhedral coneallows us to decompose the functional ϕ f in terms of the faces of the cone C.ϕ f (C) = E [ f ( ‖u‖ 2 ,‖w‖ 2 )] =d∑∑k=0 F ∈F k (C)E [ f ( ‖u‖ 2 ,‖w‖ 2 ) · 1 relint(F ) (u) ] (8.2)□
STEINER FORMULAS FOR CONES AND INTRINSIC VOLUMES 19where F k (C) is the set of k-dimensional faces of C and 1 A is the 0–1 indicator function of a Borel set A.We need to find an alternative expression for the expectation remaining in (8.2). Fix a k-dimensional faceF of C with normal face N F . The orthogonal decomposition (7.6) of the inverse image Π −1 ( )C relint(F ) impliesthat we can integrate over F and N F independently.E [ f ( ‖u‖ 2 ,‖w‖ 2 ) · 1 relint(F ) (u) ] ∫1=(2π)∫relint(F d/2 dx dy · f ( ‖x‖ 2 ,‖y‖ 2 ) · e −(‖x‖2 +‖y‖ 2 )/2 .) N FThis identity relies on the Pythagorean relation (7.4). The volume elements dx and dy derive from theLebesgue measures on lin(F ) and lin(N F ). Some care is required for the face F = {0}, in which case dx is theDirac measure at the origin; a similar issue arises when N F = {0}.To continue, we convert each of the integrals to polar coordinates [Fol99, Thm. 2.49]. This step givesE [ f ( ‖u‖ 2 ,‖w‖ 2 ) · 1 relint(F ) (u) ] )=d ¯σ k−1 d ¯σ d−k−1 · I f (k,d)(∫relint(F )∩S)(∫N k−1 F ∩S d−k−1where ¯σ j −1 denotes the uniform measure on the sphere S j −1 . The quantity I f (k,d) depends only on thefunction f and the two indices k and d. In view of the identity (7.8) for the solid angle of a cone, wedetermine thatE [ f ( ‖u‖ 2 ,‖w‖ 2 ) · 1 relint(F ) (u) ] = ∠(F )∠(N F ) · I f (k,d). (8.3)We have employed the fact that the solid angle of a cone coincides with the solid angle of its relative interior.The geometry of the face F only enters this expression through the presence of the solid angles.We are almost done now. Combine the decomposition (8.2) and the identity (8.3) to reachϕ f (C) =d∑k=0(∑)I f (k,d) ·F ∈F k (C) ∠(F )∠(N F ) =d∑I f (k,d) · v k (C). (8.4)The second relation follows from Fact 7.2, which expresses the intrinsic volumes in terms of the internal andexternal angles of the cone C. Finally, we must identify the coefficients I f (k,d). Recall that a j-dimensionalsubspace L j of R d is a polyhedral cone with v j (L j ) = 1 and v k (L j ) = 0 for k ≠ j. Applying the formula (8.4) tothe subspace L j , we learn thatϕ f (L j ) = I f (j,d) for j = 0,1,2,...,d.Substitute these identities into (8.4) to complete the proof of (8.1).8.2. Continuity of intrinsic volumes. To carry out our plan, we need to verify that the conic intrinsicvolumes of C are well defined and continuous with respect to the conic Hausdorff metric.Proposition 8.2 (Intrinsic volumes of convex cones). Consider a closed convex cone C ∈ C d .(1) Well-definition. There is a sequence (C i ) i∈N of polyhedral cones in C d that converges to C in the conicHausdorff metric. For each index k, the limit lim i→∞ v k (C i ) exists, and it is independent of the sequenceof polyhedral cones. Therefore, we may definek=0v k (C) := limi→∞v k (C i ) for k = 0,1,2,...,d. (8.5)(2) Continuity. Let (C i ) i∈N be any sequence of cones in C d that converges to C in the conic Hausdorff metric.Thenlim v k(C i ) = v k (C) for k = 0,1,2,...,d.i→∞Proposition 8.2 is not new. For instance, it is an immediate consequence of the corresponding fact [SW08,Thm. 6.5.2(b)] about spherical intrinsic volumes. Here, we develop the result as a consequence of Lemma 8.1and the continuity of the projection map, Fact 7.5. We believe that this argument provides an attractivealternative to the standard methods. Our approach rests on the following lemma.Lemma 8.3. Let X k denote a chi-square random variable with k degrees of freedom. For each d ∈ N, there is afamily { }f 1 , f 2 , f 3 ,..., f d of bounded continuous functions on R+ with the property thatE [ f j (X k ) ] {1, j = k=0, j ≠ k.□
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