From Steiner formulas for cones to concentration of intrinsic volumes

STEINER FORMULAS FOR CONES AND INTRINSIC VOLUMES 13metric projection of a matrix X onto the cone C, we first extract the symmetric part of the matrix and thencompute the positive part [Bha97, p. 99] of the Jordan decomposition:It follows thatΠ C (X ) = Π C( 12 (X + X T ) ) = 1 2 (X + X T ) + .‖Π C (X )‖ 2 F = 1 ∥ (4X + X T ) ∥ 2+ F = 1 4 tr[( X + X T ) 2+].Let G ∈ R n×n be a matrix with independent standard normal entries. Then the matrix W n = 2 −1/2( G +G T ) ∈ R n×nsymis a member of the Gaussian orthogonal ensemble (GOE). We have‖Π C (G)‖ 2 F = 1 2 tr[ (W n ) 2 +].To invoke Proposition 4.4, we must compute the variance of this quantity.Our method is to renormalize the matrix and invoke asymptotic results for the GOE. Introduce the functionh(s) := (s) 2 + = max{s,0}2 . ThenVar [ ‖Π C (G)‖ 2 ] [ nF = Var2 · tr[( n −1/2 ) 2 ] ]W n += n24 · Var[ trh ( n −1/2 )]W n .In the limit as n → ∞, the variance of the trace can be expressed in terms of an integral against a kernelassociated with the GOE [BS10, Thm. 9.2].Var [ trh ( n −1/2 W n)]→∫ 2−2∫ 2−2h ′ (s)h ′ (t) · ρ GOE (s, t)ds dtwhere the kernel takes the form(ρ GOE (s, t) = 12π 2 log 4 − st + √ )(4 − s 2 )(4 − t 2 )4 − st − √ .(4 − s 2 )(4 − t 2 )With the assistance of a computer algebra system, we determine that the double integral equals 1 + 16/π 2 .Therefore,Var [ ‖Π C (G)‖ 2 ] n 2 (F = 1 + 16 )4 π 2 + o(n 2 ) as n → ∞.Proposition 4.4 yieldsIn particular,Var[V C ] = Var [ ‖Π C (G)‖ 2 ] n 2F − 2δ(C) =4Var[V C ]δ(C)→ 16 − 1 as n → ∞.π2 ( 16π 2 − 1 )+ o(n 2 ) as n → ∞.This ratio measures how much the intrinsic volumes are spread out relative to the size of the cone.As a point of comparison, Amelunxen & Bürgisser have computed the intrinsic volumes of the psd coneexactly using methods from differential geometry [AB12b, Thm. 3.5]. The expressions, involving Mehtaintegrals, can be evaluated for low-dimensional cones, but they have resisted asymptotic analysis.6.3. Circular cones. A circular cone C in R d with angle 0 ≤ α ≤ π/2 takes the formC = Circ d (α) := { x ∈ R d : x 1 ≥ ‖x‖cos(α) } .In particular, this family includes the second-order cone L d := Circ d (π/4). Second-order cones are also knownas Lorentz cones, and they are self-dual.With some effort, it is possible to work out the intrinsic volumes of a circular cone [Ame11, Ex. 4.4.8].Instead, we apply our techniques to compute the mean and variance of the intrinsic volume random variable.This calculation demonstrates that circular cones with a small angle saturate the variance bound fromTheorem 4.5. Afterward, we sketch an argument that small circular cones also saturate the upper tail boundfrom Corollary 4.10.