What's wrong with this solution of Tarski's circle-squaring problem? Tarski's circle-squaring problem asks whether it is possible to cut up a circle into a finite number of pieces and reassemble it into a square of the same area. Note that this is different from the problem of squaring the circle (which is about compass and straightedge constructions) and also from the Banach-Tarski paradox (which involves a change of volume). Laczkovich showed in 1990 that the answer was yes, and proved the existence of a solution with about $10^{50}$ pieces. The proof was nonconstructive.

What is the error in the following proof that the answer to the problem is no, contrary to Laczkovich? At a given stage in the process of cutting and reassembly, let $u$ be the total length of all convex edges that are circular arcs whose radius equals the radius $r$ of the original circle, and similarly let $v$ be the total length of all such concave edges. Initially, $u=2\pi r$ and $v=0$. At the end of the construction, we would have to have $u=v=0$. But $u-v$ is conserved by both cutting and reassembly, so this is impossible.

[EDIT] Edited to remove mistake about Banach-Tarski.
According to the Wikipedia article you link to, the pieces in Laczkovich's decomposition are non-measurable and do not have Jordan curve boundary, so it is impossible to define the length of their edges.
Like you I have a simple (and correct) proof that the answer is no.  The error in Laczkovich's decomposition is that he's not actually cutting up a circle, he's performing an imaginary operation similar to cutting but not constrained by the need for cuts to obey the "Jordan curve boundary" and therefore, basically, anything is possible.