Surveillance Points in High Dimensional Spaces

Let us consider any computer software, relying upon a large number of parameters, typically, 40 or more, used for any type of simulation. Such a tool will certainly be a very complicated software, therefore lengthy to execute, and the number of runs one may perform will be limited. One wants to perform these runs "at best", that is at specific places, called "Surveillance Points". These investigations will then be considered as representative of any situation. These Surveillance Points may be interpreted as the center of balls, covering an hypercube which (after normalization) describes all the possible configurations of parameters. Our first result shows that these balls must be extremely large. In dimension