36 I. Convex Sets at Large under translations). To construct an invariant functional, we average ωk,L over all k-dimensional subspaces L Rd. Let Gk(Rd) be the set of all k-dimensional subspaces L Rd. It is known that Gk(Rd) possesses a manifold structure (it is called the Grassmannian) and the rotationally invariant probability measure dL. Hence, for f K(Rd) we let ωk(f) = Gk(Rd) ωk,L(f) dL. In other words, ωk(f) is the average value of ωk,L(f) over all k-dimensional sub- spaces L Rd. Clearly, ωk : K(Rd) −→ R is a valuation. For a compact convex set A Rd we define wk(A) := ωk([A]). Hence wk(A) is the average volume of projections of A onto k-dimensional subspaces in Rd. The number wk(A) is called the k-th intrinsic volume of A. It satisfies properties (8.3.1)–(8.3.3) and it is homogeneous of degree k: wk(αA) = αkwk(A) for α 0. It is convenient to agree that w0(A) = χ(A) and that wd(A) = vold(A). PROBLEMS. 1. Compute the intrinsic volumes of the unit ball B = x Rd : x≤ 1 . 2∗. Let A Rd be a compact convex set with non-empty interior. Prove that the surface area of A (perimeter, if d = 2) is equal to cdwd−1(A), where cd is a constant depending on d alone. Find cd. Here is another interesting valuation. 3. Let us fix a vector c Rd. For a non-empty compact convex set A Rd, let h(A c) = max x∈A c, x (when A is fixed, the function h(A, c) : Rd −→ R is called the support function of A). Prove that there exists a valuation νc : K(Rd) −→ R such that νc([A]) = h(A c) for every non-empty convex compact set A Rd. Hint: If c = 0, let νc(f) = α∈R α χ(fα) lim −→+0 χ(fα+ ) , where is the restriction of f onto the hyperplane H = x : c, x = α . 4∗. Let K1,K2 Rd be compact convex sets such that K1 K2 is convex. Prove that (K1 K2) + (K1 K2) = K1 + K2. Hint: Note that [K1 K2] + [K1 K2] = [K1] + [K2] and use Problem 3 to conclude that h(K1 K2 c) + h(K1 K2 c) = h(K1 c) + h(K2 c) for any c Rd. Observe that h(A + B c) = h(A c) + h(B c).
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