A unique representation of polyhedral types

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

This paper is an extension of the author's lecture "Unique Determination of Polyhedral Domains and $p$-Moduli of Path Families" given at the International Conference "Metric Geometry of Surfaces and Polyhedra" dedicated to the 100th anniversary 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…

多面体赋范空间中的唯一可实现图

Uniquely realisable graphs in polyhedral normed spaces

本文研究多面体赋范空间中框架的全局刚性问题，提出了判定算法、通用框架的必要条件，并在$\ell_\infty^d$空间中获得了更强的充分条件和二维精确刻画。

Polyhedral Finsler spaces with locally unique geodesics

We study Finsler PL spaces, that is simplicial complexes glued out of simplices cut off from some normed spaces. We are interested in the class of Finsler PL spaces featuring local uniqueness of geodesics (for complexes made of Euclidean simplices, this property is equivalent to local CAT(0)). Though non-Euclidean normed spaces never satisfy CAT(0), it turns out that they share many common features. In particular, a globalization theorem holds: in a simply-connected Finsler PL space local uniqueness of geodesics implies the global one. However the situation is more delicate here: some basic co…

Minkowski-type and Alexandrov-type theorems for polyhedral herissons

Classical H.Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H.Minkowski uniqueness theorem due to A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be described as polyhedra with injective spherical image.

Polyhedral representation conversion up to symmetries

We give a short survey on computational techniques which can be used to solve the representation conversion problem for polyhedra up to symmetries. We in particular discuss decomposition methods, which reduce the problem to a number of lower dimensional subproblems. These methods have been successfully used by different authors in special contexts. Moreover, we sketch an incremental method, which is a generalization of Fourier-Motzkin elimination, and we give some ideas how symmetry can be exploited using pivots.

具有多面体类型边界的双曲3-流形

Hyperbolic 3-manifolds with boundary of polyhedral type

本文研究具有双曲内部和凸边界的紧致3-流形，证明边界上凸双曲锥度量可由弯曲度量实现且可控多面体实现唯一，并得出凸余紧度量的全局与无穷小刚性，推进Thurston猜想。