Nonatomic problems of positioning lengthy objects (Q1819353)

The problem of optimal positioning \(\phi\) :\(\Omega\to D\) of a continuous set \(\Omega\) in a given compact region D is considered here. The optimal mapping \(\phi\) minimizes, depending on \(\phi\), the average cost of communications linking the elements belonging to \(\Omega\). The search for a solution in these problems is simpler than that one in problems of optimal positioning of a finite set \(\Omega\). The problem is reduced either to a boundary value problem or to the minimization of a quadratic form on a convex compact set in the above examples. Recommendations are given for constructing efficient solution algorithms for finite positioning problems of large dimension.