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Lazarev and Lieb showed that finitely many integrable functions from the unitinterval to $\mathbb{C}$ can be simultaneously annihilated in the $L^2$ innerproduct by a smooth function to the unit circle. Here we answer a question ofLazarev and Lieb proving a generalization of their result by lower bounding theequivariant topology of the space of smooth circle-valued functions with acertain $W^{1,1}$-norm bound. Our proof uses a relaxed notion of motionplanning algorithm that instead of contractibility yields a lower bound for the$\mathbb{Z}/2$-coindex of a space.

A nonlinear Lazarev-Lieb theorem: $L^2$-orthogonality via motion planning