▁different ▁than ▁ 0 . 2 . ▁The ▁number ▁of ▁land marks ▁increases ▁by ▁at ▁most ▁ 2 ▁in ▁each ▁query . ▁In ▁particular , ▁the ▁value ▁of ▁the ▁interval ▁[ 0 , 0 . 2 ] ▁might ▁be ▁concentr ated ▁entirely ▁near ▁ 0 , ▁or ▁entirely ▁near ▁ 0 . 2 , ▁or ▁anywhere ▁in ▁between . ▁ ▁B . ▁An ▁interval ▁between ▁two ▁consecutive ▁land marks ▁of ▁partner ▁i ▁is ▁called ▁a ▁land mark - interval ▁of ▁partner ▁i , ▁When ▁the ▁algorithm ▁dec ides ▁to ▁allocate ▁a ▁piece ▁of ▁c ake ▁to ▁partner ▁i , ▁it ▁must ▁allocate ▁a ▁piece ▁whose ▁total ▁value ▁for ▁i ▁is ▁at ▁least ▁as ▁large ▁as ▁any ▁land mark - interval ▁of ▁ ▁i . ▁The ▁proof ▁is ▁by ▁contradiction : ▁suppose ▁there ▁is ▁a ▁certain ▁land mark - interval ▁J ▁whose ▁value ▁for ▁i ▁is ▁more ▁than ▁the ▁value ▁actually ▁allocated ▁to ▁i . ▁Some ▁other ▁partner , ▁say ▁j , ▁will ▁necessarily ▁receive ▁some ▁part ▁of ▁the ▁land mark - interval ▁J . ▁By ▁paragraph ▁A , ▁it ▁is ▁possible ▁that ▁all ▁the ▁value ▁of ▁interval ▁J ▁is ▁concentr ated ▁inside ▁the ▁share ▁allocated ▁to ▁partner ▁j . ▁Thus , ▁i ▁env ies ▁j ▁and ▁the ▁division ▁is ▁not ▁env y - free . ▁ ▁C . ▁Suppose ▁all ▁partners ▁answer ▁all ▁queries ▁as ▁if ▁their ▁value ▁measure ▁is ▁uniform ▁( i . e . ▁the ▁value ▁of ▁an ▁interval ▁is ▁equal ▁to ▁its ▁length ). ▁By ▁paragraph ▁B , ▁the ▁algorithm