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The Bellows conjecture. (English) Zbl 0939.52009

Let \(S\) be a triangulated oriented surface placed in Euclidean three-space (with possible self-intersections). Denote by \(\text{vol}(S)\) the generalized, signed, volume of the domain enclosed by \(S\).
The first author [Proc. Int. Congr. Math., Helsinki 1978, Vol. 1, 407-414 (1978; Zbl 0434.53042)] conjectured that whenever we perform a rigid deformation of \(S\) (a continuous flex that preserves the edge lengths of the triangulation), the volume \(\text{vol}(S)\) remains constant (he called it the bellows conjecture).
This conjecture was proven by the second author [Usp. Mat. Nauk 50, No. 2, 223-224 (1995); translation from Russ. Math. Surv. 50, No. 2, 451-452 (1995; Zbl 0857.52004)] in the case of \(S\) being homeomorphic to a sphere; this proof was then generalized by the same author for arbitrary oriented 2-manifolds [Fundam. Prikl. Mat. 2, No. 4, 1235-1246 (1996; Zbl 0904.52002)].
The present paper presents a modified version of the generalized proof.

MSC:

52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
52B70 Polyhedral manifolds
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