Lately I’ve been thinking again about the “square pegs” problem: proving that any simple closed plane curve has an inscribed square. (I’ve blogged about this before: here, here, here, here, here.) This post is just to collect some recent links that are relevant to the problem, some of which contain new results.
Jason Cantarella has a page on the problem with lots of nice pictures of inscribed squares, like the one at the bottom of this post.
Igor Pak wrote a preprint giving two elegant proofs that every simple closed piecewise-linear curve in the plane has an inscribed square. What’s more, Igor tells me about a nice generalized conjecture: if Q is a quadrilateral with a circumscribed circle, then every smooth simple closed plane curve has an inscribed quadrilateral similar to Q. Apparently this is not always true for piecewise-linear curves!
I had a nice generalization of this problem in mind, which has the advantage of being invariant under the whole group of affine-linear transformations and not just the affine-orthogonal ones: show that every simple closed plane curve has an inscribed hexagon which is an affine-linear transform of a regular hexagon. This is carried out for smooth curves in a November 2008 preprint of Vrecica and Zivaljevic. What’s more, the conjecture apparently dates back to 1972 and is due to Branko Grunbaum. I wonder whether Pak’s methods supply a nice proof in the piecewise linear case.