From Steiner formulas for cones to concentration of intrinsic volumes

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20 M. B. MCCOY AND J. A. TROPPProof. Consider the density ρ k of the chi-square random variable X k for each k = 1,2,3,...,d.ρ k (s) =12 k/2 Γ(k/2) · sk/2 e −s/2 for s ≥ 0.These functions are bounded and continuous, and they compose a linearly independent family [CL09, Chap. 5].Introduce the d-dimensional linear space P := lin { }ρ 1 ,...,ρ d equipped with the usual inner product〈f , ρ〉 :=∫ ∞0f (s)ρ(s)ds for f ,ρ ∈ P.Standard arguments [Kre89, Lem. 8.6-2] show that { } { }ρ 1 ,...,ρ d induces a biorthogonal system f1 ,..., f d ⊂ P.By construction, the functions f j identify the number of degrees of freedom in a chi-square random variable.Indeed, let X k follow the chi-square distribution with k degrees of freedom. ThenE [ f j (X k ) ] ∫ {∞11, j = k= f j (s) ·2 k/2 Γ(k/2) sk/2 e −s/2 ds = 〈f j , ρ k 〉 =0, j ≠ k.0This is the advertised result.□We can use the functions from Lemma 8.3 in combination with Lemma 8.1 to isolate the properties ofindividual intrinsic volumes.Proof of Proposition 8.2. Let C ∈ C d be a closed convex cone. Fact 7.4 implies that there is a sequence (C i ) i∈Nof polyhedral cones in C d for which C i → C. As a consequence, C ◦ i → C ◦ as well.Consider the family { }f 1 ,..., f d of functions promised by Lemma 8.3. For each index j ≥ 1, Lemma 8.1shows thatE [ (f j ‖ΠCi (g )‖ 2 )] d∑= E [ f j (X k ) ] · v k (C i ) = v j (C i ) for i ∈ N.We claim thatk=0v j (C i ) = E [ f j(‖ΠCi (g )‖ 2 )] → E [ f j(‖ΠC (g )‖ 2 )] as i → ∞. (8.6)The limit in (8.6) does not depend on the choice of sequence, so the definition (8.5) of the intrinsic volumesof C is valid for each index k ≥ 1. For the intrinsic volume v 0 , we simply note thatv 0 (C i ) = v d (C ◦ i ) → v d (C ◦ )as i → ∞as a consequence of the definition of v d . This limit is unambiguous, so v 0 is also well-defined.To justify the calculation in (8.6), we apply the dominated convergence theorem to pass the limit throughthe expectation. Fact 7.5 shows that the metric projection is continuous; the Euclidean norm and the functionsf j are also continuous. Thus, we have the pointwise limitf j(‖ΠCi (x)‖ 2 ) → f j(‖ΠC (x)‖ 2 ) as i → ∞ for x ∈ R d .The integrands are controlled by an integrable function because f j is bounded:E [∣ (∣ f j ‖ΠCi (g )‖ 2 )∣ ∣ ] ≤ sup∣ f j (s) ∣ for i ∈ N.s≥0Dominated convergence applies, which ensures that (8.6) is correct.From here, it is easy to verify the continuity of intrinsic volumes. Suppose that (C i ) i∈N is a sequence ofclosed convex cones in C d for which C i → C. For each index k ≥ 0, we can find a sequence (C ′ i ) i∈N of polyhedralcones in C d for whichdist H (C ′ i ,C i ) for i ∈ N.This point follows from the density of polyhedral cones in C d stated in Fact 7.4 and the definition (8.5) of theintrinsic volumes. Since C i → C, this construction ensures that C ′ → C. By definition of the intrinsic volumes,iv k (C ′ i ) → v k(C). But then we must conclude that v k (C i ) → v k (C).□
STEINER FORMULAS FOR CONES AND INTRINSIC VOLUMES 218.3. Extension to general convex cones. Next, let us extend the master Steiner formula, Lemma 8.1 frompolyhedral cones to closed convex cones. Our strategy is to approximate a convex cone with a sequence apolyhedral cones, apply Lemma 8.1 to each member of the sequence, and use continuity to take the limit.Lemma 8.4 (Extension to closed convex cones). Let f : R 2 + → R be a bounded continuous function, and letC ∈ C d be a closed convex cone. Then the master Steiner formula (8.1) still holds.Proof. Fact 7.4 ensures that polyhedral cones form a dense subset of C d , so there is a sequence (C i ) i∈N ofpolyhedral cones in C d for which C i → C as i → ∞. We also have the limit C ◦ i → C ◦ . Lemma 8.1 implies thatTaking the limit as i → ∞, we reachE [ f ( ‖Π Ci (g )‖ 2 , ‖Π C ◦ (g )] d∑)‖2 = ϕi f (L k ) · v k (C i ) for i ∈ N.k=0E [ f ( ‖Π C (g )‖ 2 , ‖Π C ◦(g )‖ 2 )] =d∑ϕ f (L k ) · v k (C).To justify the limit on the left-hand side, we invoke the dominated convergence theorem. This act is legalbecause f is bounded and continuous, the squared Euclidean norm is continuous, and the metric projector iscontinuous. The limit on the right-hand side follows from the continuity of intrinsic volumes expressed inProposition 8.2.□8.4. Extension to integrable functions. We are now prepared to complete the proof of the master Steinerformula that we announced in Section 3.1. All that remains is to expand the class of functions that we canconsider. The following lemma contains the outstanding claims of Theorem 3.1.Lemma 8.5 (Extension to integrable functions). Let f : R 2 + → R be a Borel measurable function, and let C ∈ C dbe a closed convex cone. Then the master Steiner formula (8.1) still holds, provided that each expectation is finite.Proof. Let us reinterpret Lemma 8.4 as a statement about measures. The Banach space C 0 (R 2 + ) consistsof bounded and continuous real-valued functions on R 2 + that tend to zero at infinity. Consider a functionh ∈ C 0 (R 2 + ), and observe that the left-hand side of (8.1) can be written asϕ h (C) = E [ h ( ‖Π C (g )‖ 2 , ‖Π C ◦(g )‖ 2 )] ∫= h(s, t)dµ(s, t)where the measure µ is defined for each Borel set A ⊂ R 2 + by the rulek=0µ(A) := P {( ‖Π C (g )‖ 2 , ‖Π C ◦(g )‖ 2 ) ∈ A } .Similarly, the right-hand side of (8.1) can be written asd∑d∑(∫ϕ f (L k ) · v k (C) =k=0where the measure µ k is defined viak=0)h(s, t)dµ k (s, t) · v k (C)µ k (A) := P {( ‖Π Lk (g )‖ 2 , ‖Π Lk◦(g )‖ 2 ) ∈ A } .As a consequence, Lemma 8.4 demonstrates that∫ ∫ ( )d∑h dµ = h d v k (C)µ kk=0for h ∈ C 0 (R 2 + ). (8.7)We claim that (8.7) guarantees the equality of measuresd∑µ = v k (C) · µ k . (8.8)k=0Because the measures are equal, it holds for each nonnegative Borel measurable function f + : R 2 + → R + that∫d∑(∫ )f + dµ = f + dµ k · v k (C).k=0We can replace f + with any Borel measurable function f : R 2 + → R, provided that all the integrals remain finite.Reinterpreted, this observation yields the conclusion.
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