We introduce PSN polytopes, whose k-skeleton is combinatorially equivalent to
that of a product of r simplices. They simultaneously generalize both
neighborly and neighborly cubical polytopes.

We construct PSN polytopes by three different methods, the most versatile of
which is an extension of Sanyal and Ziegler's "projecting deformed products"
construction to products of arbitrary simple polytopes. For general r and k,
the lowest dimension we achieve is 2k+r+1.

Using topological obstructions similar to those introduced by Sanyal to bound
the number of vertices of Minkowski sums, we show that this dimension is
minimal if we moreover require the PSN polytope to be obtained as a projection
of a polytope combinatorially equivalent to the product of r simplices, when
the sum of their dimensions is at least 2k.