Тема: Mixed volume of infinite-dimensional convex compact sets

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.
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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)).
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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 .
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