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where the inequality follows from the fact that e$_{C}$ N − ( m − 1) ≤ 1. 

We shall show that this limit can be bounded from below by a positive constant that depends on ( x$_{0}$, y$_{0}$ ) and R but not on I . Since 

a positive lower bound on expression (A.2) can be obtained: 

<img src='content_image/315670.jpg'>

<img src='content_image/315669.jpg'>

where the first inequality follows since h ( x, y ) − f ( x ) g ( y ) > 0 in A , and the sec- ond inequality follows since the minimum value is c > 0. Therefore, it follows that 1 N − m +1 ( o$_{C}$ − e$_{C}$ ) 2 e$_{C}$ converges uniformly almost surely to a positive constant greater than c ′ = c $^{2}$R$^{2}$/ 4 , 

<img src='content_image/315668.jpg'>

The partition I either contains the cell C , or a group of cells that divide C . By Jensen’s inequality, it follows that if the partition I contains a group of cells that divide C , the score is made larger, since for any partition of the cell C , C = ∪$_{l}$ C$_{l}$ , 

<img src='content_image/315667.jpg'>

