Suppose v1,,,,,,,,,,,, vk are linearly independent vectors in Rn(1 ≤ k ≤ n). Then the set Xk = conv {±v1,,,,,,,,,±vk} is called a k-crosspolytope.
a. Sketch X1 and X2.
b. Determine the number of k-faces of the 3-dimensional crosspolytope X3 for k = 0, 1, 2. What is another name for X3?
c. Determine the number of k-faces of the 4-dimensional crosspolytope X4 for k = 0, 1, 2, 3. Verify that your answer satisfies Euler's formula
d. Find a formula for fk (Xn), the number of k-faces of Xn, for 0 ≤ k ≤ n - 1.