|ANKE WALZ, Cornell University, Ithaca, New York 14853, USA|
|The Bellows conjecture in dimension four|
I will talk about the following generalized version of the Bellows Conjecture: ``If St is a flex of an orientable singular cycle in Euclidean 4-space, then the volume of St is constant.'' A flex is a continuous motion of a configuration of vertices that fixes the lengths of the edges of all triangles of the cycle.
I. Sabitov first proved this result for dimension d=3 by using resultants to show that the volume of a polyhedron in 3-space is (up to a constant factor) integral over the squares of the edgelengths of the polyhedron. R. Connelly and A. Walz modified his proof, looking at places instead of resultants. I will show that this approach can be generalized to dimension 4. This provides a proof of the Bellows Conjecture in dimension 4.