36 I. Convex Sets at Large
under translations). To construct an invariant functional, we average ωk,L over
all k-dimensional subspaces L ⊂
be the set of all k-dimensional
subspaces L ⊂
It is known that
possesses a manifold structure (it is
called the Grassmannian) and the rotationally invariant probability measure dL.
Hence, for f ∈
In other words, ωk(f) is the average value of ωk,L(f) over all k-dimensional sub-
spaces L ⊂ Rd.
Clearly, ωk :
−→ R is a valuation. For a compact convex set A ⊂
define wk(A) := ωk([A]).
Hence wk(A) is the average volume of projections of A onto k-dimensional
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) =
for α ≥ 0. It is convenient to agree that w0(A) = χ(A) and that wd(A) =
1. Compute the intrinsic volumes of the unit ball B = x ∈
: x ≤ 1 .
Let A ⊂
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
(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
α χ(fα) − lim
χ(fα+ ) ,
where fα 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 ∈
Observe that h(A + B; c) = h(A; c) + h(B; c).