Exercise 19, § 6

a) Let A be a closed convex set in E, a product of lines, that is non-compact and such that $C_A = \{0\}$ (exerc. 18). Show that if $B = \overline{A - A}$ and if M is the convex closed envelope of $A \cup (-A)$, then B and M contain lines (use exerc. 16, b) of II, p. 85). b) Let $A_1, A_2$ be two closed convex sets in E such that $A_1 + A_2$ is closed and none of $A_1, A_2, A_1 + A_2$ contain affine lines. Show that $C_{A_1 + A_2} = C_{A_1} + C_{A_2}$ (use exerc. 16, b) of II, p. 85). c) Let A be a closed convex set in E that does not contain any affine line and $M_1, ..., M_n$ closed convex sets contained in A. If B is the convex envelope of $\bigcup_i M_i$, show that $\overline{B} = B + \sum_i C_{M_i}$ and $C_B = \sum_i C_{M_i}$ (same method).