For any regular polygon in the plane, we have its associated dihedral group, and my question concerns the other direction.
Say we have some path-connected subset of $\mathbb{R}^2$, what are the hypotheses under which its group of symmetries is a dihedral group iff it is a regular polygon?
My work: I first thought that path-connectedness might be a sufficient hypothesis, but I soon thought up some obvious counterexamples; however, all of my examples are nonconvex, so I suspect that maybe convexity might be sufficient, but I'm scared to attempt a proof because I feel I'd have to do a lot of hand-waving. (this is wrong, see the prof. Myerson's answer)
Another case where I haven't been able to produce a counterexample is a bounded convex, path-connected set, such that it has only $n$ vertices (more analytic/rigorous characterisation of this property may be non-differentiability at those points). Any counterexamples/proofs would be much appreciated!