From Steiner formulas for cones to concentration of intrinsic volumes

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10 M. B. MCCOY AND J. A. TROPPProof. The argument, based on the Laplace transform method, is standard. For any ζ > 0,P { V C − δ(C) ≥ λ } ()≤ e −ζλ · Ee ζ(V C −δ(C)) ≤ e −ζλ e 2ζ − 2ζ − 1· exp· δ(C)2where we have applied the exponential moment bound (4.6) from Theorem 4.8. Minimize the right-hand sideover ζ > 0 to obtain the first branch of the maximum in (4.12). The second exponential moment bound (4.7)leads to the second branch of the maximum in (4.12). The lower tail bound (4.13) follows from the sameconsiderations. For more details about this type of proof, see [BLM13, Sec. 2.7].□To understand the content of Corollary 4.10, it helps to make some further estimates. Comparing Taylorseries, we find that ψ(u) ≥ u 2 /(2 + 2u/3). This observation leads to a weaker form of the tail bounds (4.12)and (4.13). For λ ≥ 0,P { V C − δ(C) ≥ λ } (−λ 2 )/4≤ exp(δ(C) + λ/3) ∧ (δ(C ◦ ) − λ/3)P { V C − δ(C) ≤ −λ } (−λ 2 /4≤ exp(δ(C) − λ/3) ∧ (δ(C ◦ ) + λ/3)This pair of inequalities reflects the fact that the left tail of V C exhibits faster decay than the right tail when thestatistical dimension δ(C) is small; the tail behavior is reversed when δ(C) is close to the ambient dimension.For practical purposes, it seems better to combine these estimates into a single bound:P { |V C − δ(C)| ≥ λ } (−λ 2 )/4≤ 2exp(δ(C) ∧ δ(C ◦ for λ ≥ 0. (4.14))) + λ/3This tail bound indicates that V C looks somewhat like a Gaussian variable with mean δ(C) and variance2(δ(C) ∧ δ(C ◦ )) or less. This claim is consistent with Theorem 4.5. The result (4.14) improves over [ALMT13,Thm. 6.1], and we will see an example in Section 6.3 that saturates the bound.Our analysis suggests that the intrinsic volume sequence of a convex cone C cannot exhibit very complicatedbehavior. Indeed, the statistical dimension δ(C) already tells us almost everything there is to know. Theonly large intrinsic volumes v k (C) are those where the index k is in the range δ(C) ± const · δ(C)∧ δ(C ◦ ).The consequence of this result for conic integral geometry is that a cone with statistical dimension δ(C)behaves essentially like a subspace with dimension [δ(C)]. See [ALMT13] for more support of this point andits consequences for convex optimization.).5. INTRINSIC VOLUMES OF PRODUCT CONESSuppose that C 1 ∈ C d1 and C 2 ∈ C d2 are closed convex cones. We can form another closed convex cone bytaking their direct product:C 1 ×C 2 := { (x 1 , x 2 ) ∈ R d 1+d 2: x 1 ∈ C 1 and x 2 ∈ C 2}∈ Cd1 +d 2.The probabilistic methods of the last section are well suited to the analysis of a product cone. In this section,we compute the intrinsic volumes of a product cone using these techniques. Then we identify the mean,variance, and concentration behavior of the intrinsic volume random variable of a product cone.5.1. The product rule for intrinsic volumes. The intrinsic volumes of the product cone can be derived fromthe intrinsic volumes of the two factors.Corollary 5.1 (Product rule for intrinsic volumes). Let C 1 ∈ C d1 and C 2 ∈ C d2 be closed convex cones. Theintrinsic volumes of the product cone C 1 ×C 2 satisfyv k (C 1 ×C 2 ) =∑v i (C 1 ) · v j (C 2 ) for k = 0,1,2,...,d 1 + d 2 . (5.1)i+j =kWe present a short proof of Corollary 5.1 based on the conic Wills functional (4.5). This approach echoesHadwiger’s method [Had75] for computing the Euclidean intrinsic volumes of a product of convex sets.
STEINER FORMULAS FOR CONES AND INTRINSIC VOLUMES 11Proof. Let g 1 ∈ R d 1and g 2 ∈ R d 2be independent standard normal vectors. The direct product (g 1 , g 2 ) is astandard normal vector on R d 1+d 2. For each λ > 0, the definition (4.5) of the Wills functional gives(W C1 ×C 2(λ) = λ d 1+d 2 1 − λ2E exp · dist 2 ( ) )(g 1 , g 2 ),C 1 ×C 22(= λ d 1 1 − λ2) (E exp · dist 2 (g 1 ,C 1 ) · λ d 2 1 − λ2)E exp · dist 2 (g 2 ,C 2 ) = W C1 (λ) ·W C2 (λ).22The second identity follows from the fact that the squared distance to a product cone equals the sum of thesquared distances to the factors; we have also invoked the independence of the standard normal vectors tosplit the expectation. Applying the relation (4.5) twice, we find thatW C1 ×C 2(λ) = W C1 (λ) ·W C2 (λ) =But (4.5) also shows that(d1∑λ i v i (C 1 )i=0)∑λ j v j (C 2 ))(d2j =0d 1 ∑+d 2W C1 ×C 2(λ) = λ k v k (C 1 ×C 2 ).k=0∑d 1 +d 2=k=0Comparing coefficients in these two polynomials, we arrive at the relation (5.1).λ k∑i+j =kv i (C 1 ) · v j (C 2 ).5.2. Concentration of the intrinsic volumes of a product cone. We can employ the probabilistic techniquesfrom Section 4 to collect information about the intrinsic volumes of a product cone. Let C 1 ∈ C d1 and C 2 ∈ C d2be two cones, and consider independent random variables V C1 and V C2 whose distributions are given by theintrinsic volumes of C 1 and C 2 . In view of Corollary 5.1,v k (C 1 ×C 2 ) =∑v i (C 1 ) · v j (C 2 ) = P { V C1 +V C2 = k } for k = 0,1,2,...,d 1 + d 2 .i+j =kIn other words, the intrinsic volume random variable V C1 ×C 2of the product cone has the distributionV C1 ×C 2∼ V C1 +V C2 . (5.2)This observation allows us to compute the statistical dimension of the product cone:δ(C 1 ×C 2 ) = E [ V C1 ×C 2]= δ(C1 ) + δ(C 2 ). (5.3)Of course, we can also derive (5.3) directly from Proposition 4.3. A more interesting consequence is thefollowing expression for the variance of the intrinsic volumes:Var [ V C1 ×C 2]= Var[VC1 ] + Var[V C2 ] ≤ 2 [( δ(C 1 ) ∧ δ(C ◦ 1 )) + ( δ(C 2 ) ∧ δ(C ◦ 2 ))] . (5.4)The inequality follows from Theorem 4.5. With some additional effort, we can develop a concentration resultfor the intrinsic volumes of a product cone that matches the variance bound (5.4).Corollary 5.2 (Concentration of intrinsic volumes for a product cone). Let C 1 ∈ C d1 and C 2 ∈ C d2 be closedconvex cones. For each λ ≥ 0,P {∣ ∣ VC1 ×C 2− δ(C 1 ×C 2 ) ∣ (} −λ 2 )/4≥ λ ≤ 2 expσ 2 where σ 2 := ( δ(C 1 ) ∧ δ(C1 ◦ + λ/3)) + ( δ(C 2 ) ∧ δ(C2 ◦ )) .This represents a significant improvement over the simple tail bound from [ALMT13, Lem. 7.2]. A similarresult holds for any finite product C 1 × ··· ×C r of closed convex cones.Proof. First, recall the numerical inequalitye 2ζ − 2ζ − 1≤2 1 − 2|ζ|/3ζ 2for |ζ| < 3 2 .This estimate allows us to package the two exponential moment bounds from Theorem 4.8 as(Ee ζ(V C ζ −δ(C)) 2 (δ(C) ∧ δ(C ◦ )))≤ expfor |ζ| < 31 − 2|ζ|/32 .Applying this bound twice, we learn that the exponential moments of the random variable V C1 ×C 2satisfy(Ee ζ(V C 1 ×C 2 −δ(C 1 ×C 2 )) = Ee ζ(V C 1 −δ(C 1 )) · Ee ζ(V C 2 −δ(C 2 )) ζ 2 σ 2 )≤ exp.1 − 2|ζ|/3□
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