Let K C Rd be a non-empty convex compact set and dim K be the dimension of K (that is, the dimension of the smallest affine subspace containing K). One of the most important geometric characteristics of K is its intrinsic volumes V0(K),..., Vd(K), which are defined as the coefficients in the Steiner formula (see, e.g.,)
d
Nod(K + ABd) = Kd-kVk(K)Ad-k, A > 0, (1)
k=0
where Vold(-) denotes the volume (d-dimensional Lebesgue measure), Bk is the k- dimensional unit ball and Kk := Volk (Bk) = nk/2/Г(2 + 1) is the volume of Bk. In other words, the volume of the neighborhood is represented by a polynomial whose coefficients depend on the set K .
The intrinsic volumes play an important role in convex geometry ). In particular, that Vd(•) is the d-dimensional volume, Vd-i(’) is half the surface area for d-dimensional convex compact sets, V1(-) is the mean width, up to a constant factor, and V0(^) = 1.
Moreover, the normalization in (1) is chosen so that the intrinsic volumes of the set do not depend on the dimension of the ambient space. This means that if we embed K into RN with N ^ d, the intrinsic volumes will be the same. This observation allowed Sudakov and Chevet to generalize the concept of intrinsic volume to the case of infinite-dimensional K as follows.
Let H be an infinite-dimensional separable Hilbert space. Then for an arbitrary non-empty convex set K C H we define Vk(K), k = 0,1,..., by the formula
Vk(K)= sup Vk(K') E [0, rc], (2)
K 'CK
where the supremum is taken over all finite-dimensional convex compact subsets K of K.
In the next subsection, we formulate the well-known results demonstrating a deep connection between the intrinsic volumes of some convex compact sets and Gaussian processes.
Consider the finite-dimensional cones Cn C E0, n G N: C1 C C2 C ... C C that approximate the cone C C E0 from the inside: Cff=1Cn is dense in C. Then we have
Yj (P°s(U~=1Cn)) = Yj(C).
Proof of Statement 4. Clearly, Yj(pos^n^C^)) C Yj(C) because of „.? /П C C and monotonicity of Grassmann angles. Let us show that Yj(pos(J^=1Cn)) ^ Yj(C). By definition,
Yj (C) = sup P[C' n Wj {0}],
C'CC
where the supremum can be taken only over polyhedral finite-dimensional cones by Remark 11. Consequently, it is sufficient to prove that for every finite-dimensional polyhedral cone C' C C, d := dim C', we have
Yj(C') = PWd-j n C = {0}] C Yj(pos(J~iC„)).
The cone C' is polyhedral, hence there are unit vectors v1,... ,vl E lin C' such that C' = pos(v1,... ,vl). Consider e > 0. The set Jnf=1Cn is dense in C, hence for every vi there exists a unit vector u E Jnf=1Cn such that dH(ui,Vi) < e. Therefore, the cone C£. = pos(u1,..., ul) is close to C', i.e., dH(C£ n S(E0),C' n S(E0)) < e, where S(E0) denotes the unit sphere in E0. Let ek be a sequence of positive numbers such that lim ek = 0 as k ^ w. We want to use Statement 3 to show that limk .x Yj (C1) = Yj (C'). To do this formally we need to place all C1 into one common finite-dimensional space with C'.
Notice that dimC£ C l for every e, hence dimlin(C' J C') C d + l. This means that for every e there exist a linear subspace U£ C E0 with dim U£ = d +1 such that C' J C' C U£. Decompose U£ into the orthogonal sum U£ = lin C' ф V£. Now fix (d + l)-dimensional space Rd+l = Rd ф Rl and the isometry I between lin C' and Rd. Let J£: U£ ^ Rd+l be an isometric operator such that J£ coincides with I on lin C'. Then Yj(C') = Yj(IC'), Yj(C) = Yj(J£C£) and dH(C£ n ), C' n S(E0)) = dH(J£C£ n Sd+l-1, IC' n Sd+l-1). Therefore, J£kC£k ^ IC', as k ^ w, Statement 3 applies and
Yj (C') = Yj (IC') = lim Yj (J£k C£k) = lim Yj (C£k).
к^ж k k^-ж k
To conclude the proof notice that for every e > 0, we have C£ C pos(jn=1Cn), hence Yj(C£) C Yj(pos(j~=1Cn)).
□
We return to the proof of Theorem 7.
For cones C such that dimC < w, this theorem is proved in [7, Theorem 3.5]. It remains to check the case dim C = w. We approximate the cone C from the inside by finite-dimensional cones Cn, n E N: C1 C C2 C ... C C; Jn=1Cn is dense in C.
We will use the same argument as applied above to the mixed volumes (see Subsection 2.1 and Section 5).
The proof of Corollary 1 does not use the compactness of the set K, the key property in the proof is the existence of a natural modification of the process (0,x) on K. Therefore, by Theorem 11 we can repeat the proof of Corollary 1 for a convex GBa-cones C.
Thus, using Corollary 1, we get that un=1Spec(x1,..., xk|Cn) is almost surely dense in Spec(x1,... ,xk|C) = {((0, x1),..., (0, xk)) : 0 G C}. Hence,
E[Yj(SpecfcC)] = E[7j(pos(u~=iSpecfcCn))] = E[lim Yj(SpecfcGJ] п^ж
= lim E[Yj(SpeckCn)] = lim Yj(Cn) = Yj(pos(u~=iGn)) = Yj(C). n^A^o n^^
Here, the first and last equalities hold by Statement 4, the second and fifth by definition of Grassmann angle (15). In the third equality we have used Lebesgue’s dominated convergence theorem. Finally, the fourth equality is the assertion of the theorem for finite-dimensional cones Cn .
