Exercise 17, § 5

In $\mathbf{R}^n$, let A be a symmetric convex set of which 0 is an interior point and of which the frontier does not contain any genuine segment. Let H be a homogeneous hyperplane and D a line complementary to H. Show that there exists a point $a \in H \cap A$ such that at $a$ there is a hyperplane of support to A that is parallel to D.