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Reflection groups and the pizza theorem

The classical theorem about cutting a round pizza into 8 pieces with straight cuts passing through an arbitrary internal point and forming angles of 45 degrees says that the total areas of odd and even pieces are equal if those pieces are ordered around the center of cutting. The current paper proposes a generalization of the Pizza theorem to any dimension and discovers a relationship with the finite reflection group of the series 

 
 
 B
 n
 
 B_n
 

.