TITLE: The difference between an affine k-simplex and a rectilinear k-simplex
QUESTION [9 upvotes]: The notion of rectilinear k-simplex appears in Theorem 10.27 of Rudin's book "Principles of Mathematical analysis", then what is the definition of a rectilinear k-simplex?  I read the proof of Theorem 10.27 and think that the proof treats oriented affine k-simplex instead of rectilinear k-simplex, is it right?

REPLY [2 votes]: Yeah, I understand it the same way. Since Rudin refers back to equation (75), the definition of "oriented affine k-simplex", I believe he means "oriented affine k-simplex" by "oriented rectilinear k-simplex".