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This paper is an extension of the author's lecture "Unique Determination ofPolyhedral Domains and $p$-Moduli of Path Families" given at the InternationalConference "Metric Geometry of Surfaces and Polyhedra" dedicated to the 100thanniversary of Prof. Nikolay Vladimirovich Efimov, which was held in Moscow(Russia) in August 2010 (in this connection, see, for example [A. P. Kopylov,Unique determination of polyhedral domains and $p$-moduli of path families, In:Contemporary problems of Mathematics and Mechanics. VI. Mathematics. Issue 3.Moscow, Publishing House of Moscow university, P. 25-41 (2011)]). We expose newresults on the problem of the unique determination of conformal type fordomains in $\mathbb R^n$. It is in particular established that a (generallyspeaking) nonconvex bounded polyhedral domain in $\mathbb R^n$ ($n \ge 4$)whose boundary is an $(n - 1)$-dimensional connected manifold of class $C^0$without boundary and can be represented as a finite union of pairwisenonoverlapping $(n-1)$-dimensional cells is uniquely determined by the relativeconformal moduli of its boundary condensers.  Results on the unique determination (of polyhedral domains) of isometric typeare also obtained. In contrast to the classical case, these results present anew approach in which the notion of the $p$-modulus of path families is used.

Unique Determination of Polyhedral Domains in $\mathbb R^n$ ($n \ge 4$) and $p$-Moduli of Path Families