On partition of special 3-dimensional polyhedrons into \(d\)-convex parts
A subset \(M\) of a space with metric \(d\) is called \(d\)-convex if every \(d\)-segment with endpoints in \(M\) is contained in \(M\); here, a \(d\)-segment with endpoints \(a\) and \(b\) consists of all points \(x\) such that \(d(a,x)+ d(x,b)= d(a,b)\). The author considers real 3-space \(\mathbb{E}^3\) equipped with the metric \(d\) associated with the norm \(\|x\|=|x_1|+|x_2|+|x_3|\). In this space, a compact \(d\)-convex body is necessarily a parallelepiped with edges parallel to the coordinate axes.NEWLINENEWLINENEWLINEFor certain types of polyhedra \(P\) in \(\mathbb{E}^3\), the author obtains an explicit formula for the smallest number of \(d\)-convex parts (parallelepipeds with edges parallel to the coordinate axes) into which \(P\) can be partitioned. The polyhedra \(P\) are handlebodies, or handlebodies with holes, and they have their edges parallel to the coordinate axes.
scientific article; zbMATH DE number 1663585