▁ ▁Add itive ▁value ▁functions , ▁multi - dimensional ▁simple x ▁c ake , ▁and ▁the ▁pieces ▁must ▁be ▁simple xes . ▁ ▁The ▁proof ▁uses ▁S per ner ' s ▁theorem , ▁the ▁K - K - M ▁lemma , ▁G ale ' s ▁covering ▁lemma ▁and ▁Ky ▁F an ' s ▁lemma ▁on ▁coinc idence ▁points . ▁▁ ▁Non - add itive ▁value ▁functions , ▁multi - dimensional ▁simple x ▁c ake , ▁and ▁the ▁pieces ▁must ▁be ▁poly top es . ▁ ▁D ivid ing ▁a ▁' bad ' ▁c ake ▁▁ ▁In ▁some ▁cases , ▁the ▁' ca ke ' ▁to ▁be ▁divided ▁has ▁a ▁negative ▁value . ▁For ▁example , ▁it ▁might ▁be ▁piece ▁of ▁la wn ▁that ▁has ▁to ▁be ▁m owed , ▁or ▁a ▁piece ▁of ▁was t eland ▁that ▁has ▁to ▁be ▁clean ed . ▁Then , ▁the ▁c ake ▁is ▁a ▁' h eter ogeneous ▁bad ' ▁rather ▁than ▁a ▁' h eter ogeneous ▁good '. ▁ ▁Some ▁procedures ▁for ▁env y - free ▁c ake - cut ting ▁can ▁be ▁adapted ▁to ▁work ▁for ▁a ▁bad ▁c ake , ▁but ▁the ▁adaptation ▁is ▁often ▁not ▁trivial . ▁See ▁env y - free ▁ch ore ▁division ▁for ▁more ▁details . ▁ ▁Sum mary ▁tables ▁▁ ▁Sum mary ▁by ▁number ▁of ▁agents ▁and ▁type ▁of ▁pieces : ▁ ▁See ▁also ▁▁ ▁En vy - free ▁ch ore ▁division ▁ ▁En vy - free ▁item ▁assignment ▁ ▁Fair ▁pie - cut ting ▁ ▁Group - env