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.
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1$\begingroup$ The title is a bit misleading. (And it suggests that the answer is "yes"...) Maybe something like "Does the boundary of a convex body contain a regular planar pentagon?" $\endgroup$Joonas Ilmavirta– Joonas Ilmavirta2014-11-13 20:26:41 +00:00Commented Nov 13, 2014 at 20:26
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$\begingroup$ @ Joonas Ilmavirta: Thank you. I have edited the title up to your suggestion. $\endgroup$user64494– user644942014-11-13 20:31:19 +00:00Commented Nov 13, 2014 at 20:31
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$\begingroup$ Does a convex body need to have nonempty other interior? Otherwise the answer is no. $\endgroup$Dirk– Dirk2014-11-13 21:42:36 +00:00Commented Nov 13, 2014 at 21:42
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2$\begingroup$ If so, perhaps this can be used together with some intermediate value argument. That is, take the quadrilateral made of 4 points of a regular pentagon, then it can be inscribed in any cross section. In some, the phantom fifth point falls outside the section and in some inside, but by IV it also sometimes falls on the boundary. $\endgroup$Yoav Kallus– Yoav Kallus2014-11-14 00:34:01 +00:00Commented Nov 14, 2014 at 0:34
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1$\begingroup$ @Yoav Kallus: No non-square rhombus can be inscribed in a circle. Perhaps cyclic quadrilaterals have the property you need. $\endgroup$Douglas Zare– Douglas Zare2014-11-14 15:32:45 +00:00Commented Nov 14, 2014 at 15:32
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1 Answer
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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:
Corollary. Each convex body $K\subset \mathbb{R}^3$ is circumscribed about an affine-regular pentagon whose vertices lie on planes of support of $K$ parallel to one line.
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.
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4$\begingroup$ The recent article mathnet.ru/php/… by V. Matveev almost answers it. $\endgroup$user64494– user644942014-11-14 06:52:49 +00:00Commented Nov 14, 2014 at 6:52