  We prove a positivity result for interpolation polynomials that was
conjectured by Knop and Sahi. These polynomials were first introduced by Sahi
in the context of the Capelli eigenvalue problem for Jordan algebras, and were
later shown to be related to Jack polynomials by Knop-Sahi and
Okounkov-Olshanski. The positivity result proved here is an inhomogeneous
generalization of Macdonald's positivity conjecture for Jack polynomials. We
also formulate and prove the non-symmetric version of the Knop-Sahi conjecture,
and in fact we deduce everything from an even stronger positivity result. This
last result concerns certain inhomogeneous analogues of ordinary monomials that
we call bar monomials. Their positivity involves in an essential way a new
partial order on compositions that we call the bar order, and a new operation
that we call a glissade.
