  We consider probability measures supported on a finite discrete interval
$[0,n]$. We introduce a new finitedifference operator $\nabla_n$, defined as a
linear combination of left and right finite differences. We show that this
operator $\nabla_n$ plays a key role in a new Poincar\'e (spectral gap)
inequality with respect to binomial weights, with the orthogonal Krawtchouk
polynomials acting as eigenfunctions of the relevant operator. We briefly
discuss the relationship of this operator to the problem of optimal transport
of probability measures.
