Estimation of the Randomness of Figure Overlap

The problems considered in the paper arose as grounds for one of the methods of determining to what extent a configuration of biological objects (cells) observed in a field of vision is random. Estimates for the variance of the intersection area for cells of a given shape randomly placed in a field of vision are presented. For this, several geometric theorems of isoperimetric type are proved. Estimates for arbitrary moments of figure-intersection areas are also obtained. Based on them, the central limit theorem is proved under some additional assumptions (figures are assumed to be small and round enough).