We solve an open problem of Kushilevitz and Nisan
(1997) in communication complexity. Let $R_{eps}(f)$
and $D^{mu}_{eps}(f)$ denote the randomized and
$mu$-distributional communication complexities of
f, respectively ($eps$ a small constant). Yao's
well-known Minimax Principle states that
R_{eps}(f) = max_{mu} { D^{mu}_{eps}(f) }.
Kushilevitz and Nisan (1997) ask whether this equality
is approximately preserved if the maximization is
taken over product distributions only, rather than
all distributions $mu$:
R_{eps}(f) = (max_{mu product} { D^{mu}_{eps}(f) })^{O(1)} ?

We refute this hypothesis in the strongest terms.
Namely, we show the existence of a function
f:{-1,1}x{-1,1}->{-1,1} for which
$max_{mu product} { D^{mu}_{eps}(f) } = O(1)$
but
$R_{1/3}(f) =Omega(n).$
Furthermore, f has discrepancy $O(2^{-n(1/2-eps)}),$
which is almost the smallest possible. In particular,
f is a hardest function for every major model of
communication. Yet, the distributional method
restricted to product distributions can certify at
best an $Omega(1)$ communication lower bound for f.

Our result also gives an essentially optimal separation,
$Omega(1)$ vs. $O(2^{-n(1/2-eps)}),$ between discrepancy
under product and nonproduct distributions, improving
on the author's recent result (Sherstov 2007).
Finally, we give an essentially optimal separation,
$O(1)$ vs. $Omega(N^{1-eps}),$ between the
statistical-query complexity and sign rank of an
N-by-N sign matrix. This settles a open question
recently posed by the author (Sherstov 2007) and
completes the taxonomy of the main complexity measures
of sign matrices.