▁R ig or ously , ▁a ▁zone ▁diagram ▁is ▁any ▁solution ▁of ▁this ▁system , ▁if ▁such ▁a ▁solution ▁exists . ▁This ▁definition ▁can ▁be ▁extended ▁without ▁essentially ▁any ▁change ▁to ▁higher ▁dimensions , ▁to ▁sites ▁which ▁are ▁not ▁necessarily ▁points , ▁to ▁infinitely ▁many ▁sites , ▁etc . ▁ ▁Inter pret ation ▁In ▁some ▁settings , ▁such ▁as ▁the ▁one ▁described ▁above , ▁a ▁zone ▁diagram ▁can ▁be ▁interpreted ▁as ▁a ▁certain ▁equilibrium ▁between ▁mut ually ▁host ile ▁kingdom s ,. ▁In ▁a ▁discrete ▁setting ▁it ▁can ▁be ▁interpreted ▁as ▁a ▁stable ▁configuration ▁in ▁a ▁certain ▁combin atorial ▁game . ▁ ▁F ormal ▁definition ▁Let ▁ ▁be ▁a ▁metric ▁space ▁and ▁let ▁ ▁be ▁a ▁set ▁of ▁at ▁least ▁ 2 ▁elements ▁( ind ices ), ▁possibly ▁infinite . ▁Given ▁a ▁tuple ▁ ▁of ▁non empty ▁subsets ▁of ▁, ▁called ▁the ▁sites , ▁a ▁zone ▁diagram ▁ ▁with ▁respect ▁to ▁this ▁tuple ▁is ▁a ▁tuple ▁ ▁of ▁subsets ▁of ▁ ▁such ▁that ▁for ▁all ▁ ▁the ▁following ▁equation ▁is ▁satisfied : ▁ ▁Zone ▁diagram ▁as ▁a ▁fixed ▁point ▁The ▁system ▁of ▁equations ▁which ▁defines ▁the ▁zone ▁diagram ▁can ▁be ▁represented ▁as ▁a ▁fixed ▁point ▁of ▁a ▁function ▁defined ▁on ▁a ▁product ▁space . ▁Indeed , ▁for ▁each ▁index ▁ ▁let ▁ ▁be ▁the ▁set ▁of ▁all ▁non empty ▁subsets ▁of ▁. ▁Let ▁ ▁and ▁let ▁ ▁be ▁the ▁function ▁defined ▁by ▁, ▁where ▁ ▁and ▁ ▁Then ▁▁ ▁is ▁a ▁zone ▁diagram ▁if ▁and ▁only ▁if ▁it ▁is ▁a ▁fixed ▁point ▁of