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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. Assune 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.

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# A problem on convex geometry

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. Assune $n$ is even. Does there exist some $K$ 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.