2. Carath´ eodory’s Theorem 9

PROBLEMS.

1. Prove that the set

Δ = (ξ1, . . . , ξd+1) ∈

Rd+1

: ξ1 + . . . + ξd+1 = 1 and

ξi ≥ 0 for i = 1, . . . , d + 1

is a polytope in Rd+1. This polytope is called the standard d-dimensional simplex.

2. Prove that the set

I = (ξ1, . . . , ξd) ∈

Rd

: 0 ≤ ξi ≤ 1 for i = 1, . . . , d

is a polytope. This polytope is called a d-dimensional cube.

3. Prove that the set

O = (ξ1, . . . , ξd) ∈

Rd

: |ξ1| + . . . + |ξd| ≤ 1

is a polytope. This polytope is called a (hyper)octahedron or crosspolytope.

simplex

cube octahedron (crosspolytope)

Figure 3. Some 3-dimensional polytopes: simplex (tetrahedron), cube

and octahedron

4. Prove that the disc B = (ξ1, ξ2) ∈ R2 : ξ1 2 + ξ2 2 ≤ 1 is not a polytope.

5. Let V = C[0, 1] be the space of all real-valued continuous functions on the

interval [0, 1] and let A = f ∈ V : 0 ≤ f(τ) ≤ 1 for all τ ∈ [0, 1] . Prove that A

is not a polytope.

The following two problems constitute the Weyl-Minkowski Theorem.

6∗. Prove that a polytope P ⊂ Rd is also a polyhedron.