▁ ▁the ▁set ▁ ▁of ▁all ▁points ▁in ▁the ▁plane ▁whose ▁distance ▁to ▁the ▁given ▁site ▁ ▁is ▁not ▁greater ▁to ▁their ▁distance ▁to ▁any ▁other ▁site ▁. ▁The ▁collection ▁ ▁of ▁these ▁regions ▁is ▁the ▁Vor ono i ▁diagram ▁associated ▁with ▁these ▁sites , ▁and ▁it ▁indu ces ▁a ▁decomposition ▁of ▁the ▁plane ▁into ▁regions : ▁the ▁Vor ono i ▁regions ▁( Vor ono i ▁cells ). ▁ ▁In ▁a ▁zone ▁diagram ▁ ▁the ▁region ▁associated ▁with ▁the ▁site ▁ ▁is ▁defined ▁a ▁little ▁bit ▁differently : ▁instead ▁of ▁associ ating ▁it ▁the ▁set ▁of ▁all ▁points ▁whose ▁distance ▁to ▁ ▁is ▁not ▁greater ▁than ▁their ▁distance ▁to ▁the ▁other ▁sites , ▁we ▁associate ▁to ▁ ▁the ▁set ▁ ▁of ▁all ▁points ▁in ▁the ▁plane ▁whose ▁distance ▁to ▁ ▁is ▁not ▁greater ▁than ▁their ▁distance ▁to ▁any ▁other ▁region . ▁Form ally , ▁ ▁. ▁Here ▁ ▁denotes ▁the ▁e uclidean ▁distance ▁between ▁the ▁points ▁ ▁and ▁ ▁and ▁ ▁is ▁the ▁distance ▁between ▁the ▁point ▁ ▁and ▁the ▁set ▁. ▁In ▁addition , ▁▁ ▁since ▁we ▁consider ▁the ▁plane . ▁The ▁tuple ▁ ▁is ▁the ▁zone ▁diagram ▁associated ▁with ▁the ▁sites . ▁ ▁The ▁problem ▁with ▁this ▁definition ▁is ▁that ▁it ▁seems ▁circular : ▁ ▁in ▁order ▁to ▁know ▁ ▁we ▁should ▁know ▁ ▁for ▁each ▁index ▁ ▁but ▁each ▁such ▁ ▁is ▁defined ▁in ▁terms ▁of ▁. ▁On ▁a ▁second ▁thought , ▁we ▁see ▁that ▁actually ▁the ▁tuple ▁ ▁is ▁a ▁solution ▁of ▁the ▁following ▁system ▁of ▁equations : ▁