Here is a problem in contact topology that was suggested by Petya's answer to [this mathoverflow question](https://mathoverflow.net/questions/116200/a-problem-on-convex-geometry) of mine.


Let $S^\* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. I will think of contact elements as pairs of point-cooriented hyperplane, where the point lies on the hyperplane. **It will then make sense to say that two elements of $S^\* \mathbb{R}^n$ are parallel.**


**Question.** Let $i: S^{n-1} \rightarrow S^\* \mathbb{R}^n$ be a Legendrian embedding that is Legendrian isotopic to the manifold of all cooriented hyperplanes passing through a point. Does there necessarily exist a point $x \in S^{n-1}$ such that the contact elements $i(x)$ and 
$i(-x)$ are parallel?


*Remark.* Because Petya's proof is a simple application of critical point theory for the support function and the support function is just a simple instance of a generating function for a Legendrian submanifold, I'm guessing off the top of my head that his proof extends. The condition of being Legendrian isotopic to the manifold of all cooriented hyperplanes passing through a point garantees the existence of a generating function quadratic at infinity.



From convex geometry to contact topology
This is true for mere topological reasons: By a result due to G. Hirsch, every self-map of a sphere of odd degree sends a pair of antipodal points to a pair of antipodal points. See e.g. Theorem 6.2 in §9 of [A. Granas & J. Dugundji; *Fixed Point Theory*, Springer (2003)]


