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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