  A 0/1-simplex is the convex hull of n+1 affinely independent vertices of the
unit n-cube I^n. It is nonobtuse if none its dihedral angles is obtuse, and
acute if additionally none of them is right. Acute 0/1-simplices in I^n can be
represented by 0/1-matrices P of size n x n whose Gramians have an inverse that
is strictly diagonally dominant, with negative off-diagonal entries.
  In this paper, we will prove that the positive part D of the transposed
inverse of P is doubly stochastic and has the same support as P. The negated
negative part C of P^-T is strictly row-substochastic and its support is
complementary to that of D, showing that P^-T=D-C has no zero entries and has
positive row sums. As a consequence, for each facet F of an acute 0/1-facet S
there exists at most one other acute 0/1-simplex T in I^n having F as a facet.
We call T the acute neighbor of S at F.
  If P represents a 0/1-simplex that is merely nonobtuse, P^-T can have entries
equal to zero. Its positive part D is still doubly stochastic, but its support
may be strictly contained in the support of P. This allows P to be partly
decomposable. In theory, this might cause a nonobtuse 0/1-simplex S to have
several nonobtuse neighbors at each of its facets.
  Next, we study nonobtuse 0/1-simplices S having a partly decomposable matrix
representation P. We prove that such a simplex also has a block diagonal matrix
representation with at least two diagonal blocks, and show that a nonobtuse
simplex with partly decomposable matrix representation can be split in mutually
orthogonal fully indecomposable simplicial facets whose dimensions add up to n.
Using this insight, we are able to extend the one neighbor theorem for acute
simplices to a larger class of nonobtuse simplices.
