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.
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