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 : ξ2 1 + 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.

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