Appendix C - Unsolved Problems

Summary

The following is a list of unsolved problems in the area of paradoxical decompositions, equidecomposability, and finitely additive measures. The order represents the author's view as to their interest and importance.

Marczewski's Problem, Circa 1930 (3.12, 9.9)

Is there a finitely additive, isometry-invariant measure on the Borel sets in Sⁿ, n ≥ 2 (or Rⁿ, n ≥ 3), that has total measure one (or, normalizes the unit cube) and vanishes on meager sets? Such a measure cannot be countably additive (9.15), and so by 13.5, it is consistent with ZF + DC that no such finitely additive Borel measure exists. An equivalent problem in the case of R³ is: Is the unit cube paradoxical using pieces that have the Property of Baire?

Tarski's Circle-Squaring Problem, 1924 (7.5)

Is a circle (with interior) in the plane equidecomposable to a square (necessarily of the same area)?

Variations

1. (a) (p. 102) Can a negative solution be obtained if the pieces are restricted to the Borel sets? It is known that restriction to pieces that are parts of Jordan curves or two-cells (interior of a Jordan curve) yields a negative solution.

2. (b) (3.14) Is a regular tetrahedron in R³ equidecomposable to a cube using measurable pieces? A restriction to polyhedral pieces yields a negative solution (Hilbert's Third Problem).

3. […]