▁value ▁density ▁functions . ▁ ▁Hard ness ▁result ▁▁ ▁Every ▁env y - free ▁procedure ▁for ▁n ▁people ▁requires ▁at ▁least ▁ Ω ( n 2 ) ▁queries . ▁The ▁proof ▁re lies ▁on ▁a ▁careful ▁analysis ▁of ▁the ▁amount ▁of ▁information ▁the ▁algorithm ▁has ▁on ▁each ▁partner . ▁ ▁A . ▁Assume ▁that ▁the ▁c ake ▁is ▁the ▁ 1 - dimensional ▁interval ▁[ 0 , 1 ], ▁and ▁that ▁the ▁value ▁of ▁the ▁entire ▁c ake ▁for ▁each ▁of ▁the ▁partners ▁is ▁normal ized ▁ 1 . ▁ ▁In ▁each ▁step , ▁the ▁algorithm ▁asks ▁a ▁certain ▁partner ▁either ▁to ▁evaluate ▁a ▁certain ▁interval ▁contained ▁in ▁[ 0 , 1 ], ▁or ▁to ▁mark ▁an ▁interval ▁with ▁a ▁specified ▁value . ▁In ▁both ▁cases , ▁the ▁algorithm ▁g athers ▁information ▁only ▁about ▁intervals ▁whose ▁end - points ▁were ▁mentioned ▁in ▁the ▁query ▁or ▁in ▁the ▁reply . ▁Let ' s ▁call ▁these ▁end points ▁land marks . ▁▁ ▁Initial ly ▁the ▁only ▁land marks ▁of ▁i ▁are ▁ 0 ▁and ▁ 1 ▁since ▁the ▁only ▁thing ▁the ▁algorithm ▁knows ▁about ▁partner ▁i ▁is ▁that ▁vi ([ 0 , 1 ]) = 1 . ▁If ▁the ▁algorithm ▁asks ▁partner ▁i ▁to ▁evaluate ▁the ▁interval ▁[ 0 . 2 , 1 ], ▁then ▁after ▁the ▁reply ▁the ▁land marks ▁of ▁i ▁are ▁{ 0 , 0 . 2 , 1 }. ▁The ▁algorithm ▁can ▁calculate ▁vi ([ 0 , 0 . 2 ]), ▁but ▁not ▁the ▁value ▁of ▁any ▁interval ▁whose ▁endpoint ▁is