From Steiner formulas for cones to concentration of intrinsic volumes

STEINER FORMULAS FOR CONES AND INTRINSIC VOLUMES 156.3.2. Tail behavior. Circular cones also exhibit tail behavior that matches the predictions of Corollary 4.10exactly. It takes some technical effort to establish this claim in detail, so we limit ourselves to a sketch of theargument. These ideas are drawn from [ALMT13, Sec. 6.2].Fix an angle 0 < α ≪ π 2 , and abbreviate q = sin2 (α). Consider the circular cone C = Circ d (α) where thedimension takes the form d = 2(n +1) where n is a large integer. In particular, the formula (6.2) shows that thestatistical dimension δ(C) ≈ d sin 2 (α) ≈ 2nq. It can be established that the odd intrinsic volumes of C follow abinomial distribution [Ame11, Ex. 4.4.8]:2 v 2k+1 (C) = P { Y = k} where Y = Y n ∼ BINOMIAL(n, q).As a consequence of the Gauss–Bonnet Theorem [SW08, Thm. 6.5.5], there is an interlacing result [ALMT13,Prop. 5.6] for the upper tail of V C .P { Y ≥ k } ≤ P { V C ≥ 2k } ≤ P { Y ≥ k − 1 } .Thus, accurate probability bounds for V C follow from bounds for the binomial random variable Y .Our tail inequality, Corollary 4.10, predicts that V C has subgaussian behavior for moderate deviations. Tosee that circular cones actually display this behavior, we turn to the classical limits for the binomial randomvariable Y n . The Laplace–de Moivre central limit theorem states thatP { Y n − nq ≥ t √ nq(1 − q) } → 1 − Φ(t)as n → ∞ with q fixed.Here, Φ denotes the distribution function of a standard normal variable. When q is small and n is large, wecan invoke the approximation δ(C) ≈ 2nq and a tail bound for the normal distribution to obtainP { V C − δ(C) ≥ λ √ δ(C) } ≈ P { V C − 2nq ≥ λ √ 2nq(1 − q) } ≈ P { Y n − nq ≥ (2 −1/2 λ) √ nq(1 − q) } ≈ e −λ2 /4 .This expression matches the behavior expressed in the weaker tail bound (4.14). In other words, we see thatthe intrinsic volumes of a small circular cone have subgaussian concentration for moderate deviations, withvariance approximately 2δ(C).Corollary 4.10 also predicts that V C has Poisson tails for very large deviations. Vanishingly small circularcones display this behavior. Suppose that q = q n = b/n for a large constant b. The approximation δ(C) ≈ 2band Chernoff’s bound for the tail of a binomial random variable together giveP { V C − δ(C) ≥ λδ(C) } ≈ P { V C − 2b ≥ 2λb } ≈ P { Y n − b ≥ λb } ≈ e b (λ−(1+λ)log(1+λ)) .After a change of variables, this formula coincides with the tail bound (4.6). The Chernoff bound is quiteaccurate in this regime, so we see that (4.6) is saturated by vanishingly small circular cones in high dimensions.6.4. Summary of calculations. We conclude this section with an overview of the statistical dimension andvariance calculations. See Table 6.1 for this material. Observe that the ratio of the variance Var[V C ] to thestatistical dimension δ(C) can range from zero to two. A subspace L k with dimension k shows that the lowerbound is achievable across the entire range of statistical dimensions. The circular cones Circ d (α) show thatthe upper bound is saturated by cones whose statistical dimension is small. Amelunxen has conjectured thatsomewhat tighter bounds are possible when δ(C) ≈ 1 2d, but this remains to be established.7. BACKGROUND ON CONIC GEOMETRYThis section summarizes the foundational material that we require to establish the master Steiner formula.We provide sketches or references in lieu of proofs to keep the presentation lean. Most of the material here isdrawn from the books [Roc70, RW98, Bar02, SW08].7.1. Basics. We work in the Euclidean space R d . This space is equipped with the Euclidean inner product〈·, ·〉, the associated norm ‖·‖, and the norm topology. We write 0 or 0 d for the origin of R d .The linear hull lin(K ) of a convex set K ⊂ R d is the intersection of all subspaces that contain K . The relativeinterior relint(K ) is the interior with respect to the relative topology induced by R d on the linear hull lin(K ).We define the distance function dist 2 (x,K ) := inf { ‖x − y‖ 2 : y ∈ K } .We write P for the probability of an event and E for the expectation operator. The symbol ∼ denotesequality of distribution. We reserve the letter g for a standard normal vector, and θ denotes a vector uniformlydistributed on the sphere. The dimensions are determined by context.