Sharing pizza among a two power number of friends

We extend the work of Hirschhorn, Hirschhorn, Hirschhorn, Hirschhorn and Hirschhorn on the problem of sharing pizza among several people.More specifically, we present a class of Coxeter arrangements in an even dimension n such that if one cuts a ball containing the origin with one of these arrangements, the slices can be partitioned into 2 n/2 blocks such that each block has the same volume.