▁of ▁env y - f reen ess . ▁ ▁When ▁the ▁pieces ▁may ▁be ▁dis connected , ▁a ▁weight ed ▁env y - free ▁division ▁always ▁exists ▁and ▁can ▁be ▁found ▁by ▁the ▁Robert son - Web b ▁protocol , ▁for ▁any ▁set ▁of ▁weights . ▁▁▁ ▁Z eng ▁presented ▁an ▁alternative ▁algorithm ▁for ▁approximate ▁weight ed ▁env y - free ▁division , ▁which ▁requires ▁a ▁smaller ▁number ▁of ▁cut s . ▁ ▁But ▁when ▁the ▁pieces ▁must ▁be ▁connected , ▁a ▁weight ed ▁env y - free ▁division ▁may ▁not ▁exist . ▁To ▁see ▁this , ▁note ▁that ▁every ▁weight ed - env y - free ▁division ▁is ▁also ▁weight ed - pro port ional ▁with ▁the ▁same ▁weight - vector ; ▁this ▁means ▁that , ▁for ▁every ▁agent ▁i ▁with ▁value ▁measure ▁Vi : It ▁is ▁known ▁that ▁weight ed - pro port ional ▁allocation ▁with ▁connected ▁pieces ▁may ▁not ▁exist : ▁see ▁proportional ▁c ake - cut ting ▁with ▁different ▁ent it lement s ▁for ▁an ▁example . ▁ ▁Note ▁that ▁there ▁is ▁an ▁alternative ▁definition ▁of ▁weight ed - env y - free ▁division , ▁where ▁the ▁weights ▁are ▁assigned ▁to ▁pieces ▁rather ▁than ▁to ▁agents . ▁With ▁this ▁definition , ▁a ▁weight ed ▁env y - free ▁division ▁is ▁known ▁to ▁exist ▁in ▁the ▁following ▁cases ▁( each ▁case ▁general izes ▁the ▁previous ▁case ): ▁▁ ▁Add itive ▁value ▁functions , ▁ 1 - dimensional ▁c ake ▁( interval ), ▁and ▁the ▁pieces ▁must ▁be ▁connected ▁intervals .