Consider a convex body $K \subset \mathbb{R}^n$ containing the origin in its interior. Although the body is not necessarily symmetric, let us say that two points in its boundary $\partial K$ are *antipodal* if the origin lies in the segment that joins them.

**Question 1.** Assuming that $n$ is odd, does there always exist a pair of parallel affine hyperplanes that support $K$ at a pair of antipodal points?

If the boundary of $K$ is $C^1$ this is true and here is the simple proof: For every point $x \in \partial K$ consider the tangent hyperplane at $x$ and the tangent hyperplane at the point $\bar{x}$ antipodal to it. Translate this second hyperplane so that it passes through $x$ and consider its intersection with $T_x \partial K$. If the hyperplanes are not parallel, we obtain an $(n-2)$-plane tangent to $\partial K$ at $x$. If the tangent hyperplanes at antipodal points are never parallel, we get a continuous field of tangent hyperplanes in the tangent bundle of $\partial K$. Since $\partial K$ is an even-dimensional sphere, this is impossible. Q.E.D.

What happens if the boundary is not smooth and we allow some points to be points of support for multiple support hyperplanes?

**Question 2.** Assume $n$ is even. Does there exist some convex body $K$ (with a possibly smooth boundary) for which the support planes at antipodal points are never parallel?

These questions came up in a conversation with Constantin Vernicos who convinced me that we didn't really need the answers for what we were doing. Still, I remain curious about the subject.