Suppose v_{1,...........,}v_k are linearly independent vectors in Rⁿ (1 ≤ k ≤ n). Then the set X^k = conv {±v₁,.......,±v_k} is called a k-cross polytope.

a. Sketch X¹ and X².

b. Determine the number of k-faces of the 3-dimensional cross polytope X³ for k = 0,1,2. What is another name for X³?

c. Determine the number of k-faces of the 4-dimensional cross polytope X⁴ for k =  0,1,2,3. Verify that your answer satisfies Euler’s formula.

d. Find a formula for f_k(Xⁿ), the number of k-faces of Xⁿ, for 0 ≤ k ≤ n – 1.