How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ includes 5 points which form a regular planar pentagon? The following consideration suggests the answer "yes": if we assume the boundary is described by the equation $f(x,y,z)=0$, then we have a system of $5+5+3=13$ equations in $15$ variables. In the general case an infinite solution set is expected. However, the convexity is not used here. The question was asked, but not answered, in [MSE](https://math.stackexchange.com/questions/1005996/does-there-exist-regular-planar-pentagon). 



Does the boundary of a convex body contain a regular planar pentagon?
A MathSciNet search turns up the following paper: V. V. Makeev, Polygons inscribed in a closed curve and in a three-dimensional convex body, *Journal of Mathematical Sciences* **161** (2009), 419–423 (translated from Russian). The final result in the paper is:



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> **Corollary**. Each convex body $K\subset \mathbb{R}^3$ is circumscribed about an [affine-regular](http://en.wikipedia.org/wiki/Affine-regular_polygon) pentagon whose vertices lie on planes of support of $K$ parallel to one line.
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This doesn't directly answer your question, but the conspicuous absence of your question from the paper perhaps suggests that it is an open problem. Regardless, this paper would seem to be a good entry point into the literature.


