Dependence of the heavily covered point on parameters.

We examine Gromov's method of selecting a point ``heavily covered'' by simplices chosen from a given finite point sets, in order to understand the dependence of the heavily covered point on parameters. There is no continuous dependence in this problem, but it is possible to utilize the ``homological continuous dependence'' of the heavily covered point, if we follow Gromov's approach to the problem. This allows us to infer some corollaries in a usual way. We also give an elementary argument to prove the simplest of these corollaries in the planar case and discuss other approaches and some open problems in the area. ; Non UBC ; Unreviewed ; Author affiliation: Moscow Institute of Physics and Technology ; Faculty