For a compact right-angled polyhedron ▫$R$▫ in Lobachevskii space ▫${\mathbb H}^3$▫, let ▫${\rm{vol}}(R)$▫ denote its volume and ▫${\rm{vert}}(R)$▫, the number of its vertices. Upper and lower bounds for ▫${\rm{vol}}(R)$▫ were recently obtained by Atkinson in terms of ${\rm{vert}}(R)$. In constructing a two-parameter family of polyhedra, we show that the asymptotic upper bound ▫$5v_3/8$▫, where ▫$v_3$▫ is the volume of the ideal regular tetrahedron in ▫${\mathbb H}^3$▫, is a double limit point for the ratios ▫${\rm{vol}}(R)/{\rm{vert}}(R)$▫. Moreover, we improve the lower bound in the case ▫${\rm{vert}}(R)\leqslant 56$▫.

COBISS.SI-ID: 15843929

We consider polyhedral approximations of strictly convex compacta in finite-dimensional Euclidean spaces (such compacta are also uniformly convex). We obtain the best possible estimates for errors of considered approximations in the Hausdorff metric. We also obtain new estimates of an approximate algorithm for finding the convex hulls.

COBISS.SI-ID: 15693913

We study algebraic and topological properties of the convolution semigroups of probability measures on a topological groups and show that a compact Clifford topological semigroup ▫$S$▫ embeds into the convolution semigroup ▫$P(G)$▫ over some topological group ▫$G$▫ if and only if ▫$S$▫ embeds into the semigroup ▫$\exp(G)$▫ of compact subsets of ▫$G$▫ if and only if ▫$S$▫ is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup ▫$S$▫ embeds into the functor-semigroup ▫$F(G)$▫ over a suitable compact topological group ▫$G$▫ for each weakly normal monadic functor ▫$F$▫ in the category of compacta such that ▫$F(G)$▫ contains a ▫$G$▫-invariant element (which is an analogue of the Haar measure on ▫$G$▫).

COBISS.SI-ID: 15950681