On unique determination of domains in Euclidean spaces (Q2519254)

The paper contains eight sections. Sections 1--7 give an overview on the results concerning to the problem of unique determination of domains of isometric type in Euclidean space. These results are obtained by the efforts of Russian mathematicians (A. D. Aleksandrov, V. A. Aleksandrov, Yu. G. Reshetnyak, Yu. A. Volkov, and others). The main result of the paper is given in the Section 8. Theorem 8.1. Assume that \(n\geq 4\). Then every bounded convex polyhedral domain \( U\) in \(\mathbb R^n\) (this is a non-empty bounded intersection of a finite set of open \(n\)-dimensional half-spaces) is uniquely determined by the relative conformal moduli of its boundary conductor in the class \(\mathcal P\) of all bounded convex polyhedral domains \(V\) in \(\mathbb R^n\). \( U\) can be determined in the class \(\mathcal P\) up to a similarity transformation, i.e., an affine conformal transformation.