Piano Mover's Problem

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

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. R. "Moving Sofa Constant." §8.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 519-523, 2003.

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.