In a microfluidic setting, I have encountered a puzzle of finding the minimum size (smallest total-length $L$) of a polymer (can be hyperbranched or loop, whatever shape that you can make from merging curves together) so that it cannot pass through a circular orifice (of radius $R$). The polymer can be assumed to be non-deformable, and all three-dimensional degrees of freedom (translation and rotation) are allowed for the polymer motion.
Does anyone have any idea where can I start? Naively, a ring-like polymer of circular shape radius $R$ (which has total-length $L=2\pi R$) can't pass through the orifice, but I don't know if it is the smallest total-length possible, and how to prove it rigorously.