 Solution

Fix any direction and choose a line $L$ in that direction lying outside the bounded domain. Steiner symmetrization in that direction preserves convexity and, by part (i), preserves area. Since $\Omega$ minimizes perimeter among admissible convex domains of that area,Part (ii) gives the reverse inequality, so equality holds. Its equality characterization says that $\Omega$ has an axis of symmetry parallel to $L$. Since the direction was arbitrary, $\Omega$ admits axes of symmetry in every direction.