▁looking ▁for ▁divisions ▁which ▁are ▁only ▁approximately - env y - free . ▁There ▁are ▁two ▁ways ▁to ▁define ▁the ▁approximation ▁- ▁in ▁units ▁of ▁length ▁or ▁in ▁units ▁of ▁value . ▁ ▁L ength - based ▁approximation ▁uses ▁the ▁following ▁definitions : ▁ ▁A ▁partition ▁of ▁a ▁c ake ▁is ▁represented ▁by ▁the ▁n ▁lengths ▁of ▁intervals ▁it ▁creates . ▁So ▁( 0 . 2 , 0 . 5 , 0 . 3 ) ▁is ▁a ▁partition ▁of ▁the ▁unit ▁interval ▁to ▁three ▁sub - interval s ▁with ▁lengths ▁ 0 . 2 , ▁ 0 . 5 ▁and ▁ 0 . 3 ▁( it ▁is ▁generated ▁by ▁cutting ▁the ▁unit ▁interval ▁at ▁ 0 . 2 ▁and ▁at ▁ 0 . 7 ). ▁ ▁A ▁multi - partition ▁is ▁a ▁set ▁of ▁several ▁different ▁partitions . ▁ ▁A ▁multi - partition ▁X ▁is ▁called ▁env y - free ▁if ▁there ▁exists ▁a ▁permut ation ▁of ▁the ▁partners ▁such ▁that , ▁for ▁every ▁i , ▁there ▁exists ▁an ▁element ▁of ▁X ▁such ▁that ▁the ▁i - th ▁partner ▁pre fers ▁the ▁i - th ▁segment . ▁ ▁A ▁ δ - multi - partition ▁is ▁a ▁multi - partition ▁in ▁which , ▁for ▁each ▁pair ▁of ▁partitions , ▁the ▁difference ▁between ▁each ▁of ▁their ▁coordinates ▁is ▁at ▁most ▁ δ . ▁For ▁example : ▁{ ( 0 . 2 + δ , 0 . 5 , 0 . 3 ), ▁( 0 . 2 , 0 . 5 + δ , 0