Suppose $A,B$ are polygons of equal area. By the [Wallace-Bolyai-Gerwien theorem](https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem), $A$ and $B$ are equidissectable: we can make finitely many straight-line cuts in $A$ and rearrange the resulting pieces via rigid motions to get a copy of $B$ (allowing pieces to overlap at their boundaries).

Define the **cost** of an equidissection of $A$ to $B$ to be the sum of the lengths of the cuts involved. Note that an equidissection with many cuts can still have a small cost, so this doesn't necessarily align with the more natural idea of minimizing the number of cuts involved.

> **Question**: Does a minimal-cost equidissection always exist?

Since the cost of a given dissection is a positive real number, there is no obvious problem with always being able to find a *slightly* cheaper equidissection. However, I have no idea how to produce a counterexample. More embarrassingly, I don't even see whether minimal-cost equidissections exist in simple cases - e.g. a square to an equal-area equilateral triangle.

I would also be interested in the analogous question for other geometries *(e.g. for the hyperbolic plane, or for polyhedra in Euclidean space of equal volume and Dehn invariant with cut surface area in place of cut length)*, but I'm primarily interested in polygons in the Euclidean plane.