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.    If all the edges of a d -simplex T have the same length, then T is regular. However, if d \geq 3 , then it is clear that the facets of T may have the same (d-1) -volume without T being regular. Here, the question of the extent to which the equality of r -volumes of the r -faces of T implies regularity of T is investigated, the case r = d-2 proving most fruitful.

Simplices with Equiareal Faces