Piano Mover's Problem

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Given an open subset $U$ in $n$-D space and two compact subsets $C_0$ and $C_1$ of $U$, where $C_1$ is derived from $C_0$ by a continuous motion, is it possible to move $C_0$ to $C_1$ while remaining entirely inside $U$?

See also Moving Ladder Constant, Moving Sofa Constant

References

Buchberger, B.; Collins, G. E.; and Kutzler, B. ``Algebraic Methods in Geometry.'' Annual Rev. Comput. Sci. 3, 85-119, 1988.

Feinberg, E. B. and Papadimitriou, C. H. ``Finding Feasible Points for a Two-point Body.'' J. Algorithms 10, 109-119, 1989.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/sofa/sofa.html

Leven, D. and Sharir, M. ``An Efficient and Simple Motion Planning Algorithm for a Ladder Moving in Two-Dimensional Space Amidst Polygonal Barriers.'' J. Algorithms 8, 192-215, 1987.