Can any triangle be inscribed in any convex figure? i.e. given a convex figure and a triangle can we transpose and scale and rotate that triangle so that its vertices are on the boundary of the convex figure?
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1$\begingroup$ You're going to need it to be convex with nonempty interior in the plane...a line is convex, but good luck inscribing triangles in it! Also, you want "inscribed" based on the last sentence. $\endgroup$– Charles SiegelCommented Aug 16, 2010 at 3:28
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A more general result is known: if $C$ is any Jordan curve and $T$ is a triangle then there exists a triangle similar to $T$ inscribed in $C.$ Moreover, the vertices of such triangles are dense in $C.$ See the references in the Wikipedia article on the Inscribed Square Problem.
answered Aug 16, 2010 at 3:32
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$\begingroup$ That is cool but I'm having trouble understanding what does "dense" mean in this context? $\endgroup$ Commented Aug 16, 2010 at 5:04
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$\begingroup$ It means that for any point $P$ on $C$ and any $\epsilon>0,$ there is an inscribed triangle similar to $T$ one of whose vertices is within $\epsilon$ of $P.$ $\endgroup$ Commented Aug 16, 2010 at 5:09
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$\begingroup$ Thanks, I can imagine how that is true by shifting a vertex of one solution and continuously adjusting the other two by a linear gradient. Still I am curious if both of the other vertices are also within epsilon so that the similar triangle T' is almost equal to T? $\endgroup$ Commented Aug 16, 2010 at 5:20