1. Give an example of a set A in R2 which is radial at every point of the interval {(x, 0): |x | 1} but such that this interval is not included in the interior of A.

2. Show that the ellipsoid x 2/a2 + y2/b2 + z2/c2 ≤ 1 is convex. Find a support plane at each point of its boundary and show that it is unique.

3. In R3, which planes through the origin are support planes of the unit cube [0, 1]3? Find all the support planes of the cube.

4. The Banach space c0 of all sequences of real numbers converging to 0 has the norm I{xn }I := supn |xn |. Show that each support hyperplane H at a point x of the boundary ∂ B of the unit ball B := {y: IyI ≤ 1} also contains other points of ∂ B. Hint: See Problem 7 of §6.1.