From Steiner formulas for cones to concentration of intrinsic volumes

STEINER FORMULAS FOR CONES AND INTRINSIC VOLUMES 17Each polyhedral cone C induces a tiling of R d as an orthogonal sum of the faces of C and the normal facesof C ◦ . The following statement of this claim amplifies an observation of McMullen [McM75, Lem. 3].Fact 7.1 (The tiling induced by a polyhedral cone). Let C ∈ C d be a polyhedral cone. Then the inverse image ofthe relative interior of a face F has the orthogonal decomposition( )relint(F ) = relint(F ) + NF . (7.6)Π −1CMoreover, the space R d is a disjoint union of the inverse images of the faces of the cone C:R d ⊔ ( )=relint(F ) + NF . (7.7)F a face of CFact 7.1 is almost obvious from the orthogonal decomposition (7.2). See [McC13, Prop. A.8] for a detailedproof.7.4. The solid angle of a cone. Let C be a convex cone whose linear hull is j-dimensional. The solid angleof the cone is defined as1∠(C) := e −‖x‖2 /2 dx = P { g C ∈ C } = P { θ C ∈ C } . (7.8)(2π) j /2 ∫CThe volume element dx derives from the Lebesgue measure on the linear hull lin(C). The random vector g Chas the standard normal distribution on lin(C), and θ C is uniformly distributed on the unit sphere in lin(C).We use the convention that the unit sphere in the zero-dimensional Euclidean space R 0 is the set S −1 := {0}.Let C be a polyhedral cone, and let F be a face of C with normal face N F . The internal angle of F is thesolid angle ∠(F ), while the external angle of F is the solid angle ∠(N F ). The intrinsic volumes of a polyhedralcone can be written in terms of the internal and external angles of the faces.Fact 7.2 (Intrinsic volumes and polyhedral angles). Let C ∈ C d be a polyhedral cone, and let F k (C) be thefamily of k-dimensional faces of C. Thenv k (C) = ∑ F ∈F k (C) ∠(F )∠(N F ).Fact 7.2 is a direct consequence of the Definition 2.1 of the intrinsic volumes of a polyhedral cone, theorthogonal decomposition (7.6) of the inverse image of a face, and the geometric interpretation (7.8) ofthe solid angles. This result can be traced at least as far back as [McM75]; see also [SW08, Eqn. (6.47)]. Acomplete proof appears in [McC13, Prop. A.8].Remark 7.3 (Alternative notation). In the literature, the internal angle of a face F of a cone C is oftendenoted by β(0,F ), and the external angle is often denoted by γ(F,C).7.5. The Hausdorff topology on convex cones. In this section, we develop a metric topology on the classC d of closed convex cones. This topology leads to notions of approximation and convergence, and it providesa way to extend results for polyhedral cones to general closed convex cones. See [Ame11, Sec. 3.2] for amore comprehensive treatment.To construct an appropriate metric, we begin by defining the angular distance between two nonzerovectors:( ) 〈x, y〉dist s (x, y) := arccosx, y ∈ R d \ {0}.‖x‖‖y‖We instate the conventions that dist s (0,0) = 0 and dist s (x,0) = dist s (0, x) = π/2 for x ≠ 0. This definition extendsto closed convex cones C,C ′ ∈ C d via the ruledist s (C,C ′ ) := inf x∈Cy∈C ′ dist s (x, y) when C,C ′ ≠ {0}.The trivial cone {0} demands special attention. We set dist s ({0},{0}) = 0, while dist s ({0},C) = dist s (C,{0}) = π/2when C ≠ {0}.The angular expansion T s (C,α) of a cone C ∈ C d by an angle 0 ≤ α ≤ 2π is the union of all rays that liewithin an angle α of the cone. Equivalently,T s (C,α) := { x ∈ R d : dist s (x, y) ≤ α for some y ∈ C } .Note that the expansion T s (C,α) of a convex cone need not be convex for any α > 0. For instance, the angularexpansion of a proper subspace is never convex.