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 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 ∈

Rd.

Observe that h(A + B; c) = h(A; c) + h(B; c).