Exercise 6.69. For a ? R<sup>n</sup> \ {0} and b ? R, the sets
H(a, b) = {x ? R<sup>n</sup> : (a,x) ? b} and P(a, b) = {x ? R<sup>n</sup> : (a,x) = b}
are called a half space and a hyperplane, respectively.
(i) Prove that H(a, b) is closed and convex.
(ii) Prove that P(a, b) is closed and convex.
(iii) Prove that P(a, b) is the boundary of H(a, b).
(iv) Find the interior of H(a, b).