  A strong direct product theorem states that, in order to solve k instances of
a problem, if we provide less than k times the resource required to compute one
instance, then the probability of overall success is exponentially small in k.
In this paper, we consider the model of two-way public-coin communication
complexity and show a strong direct product theorem for all relations in terms
of the smooth rectangle bound, introduced by Jain and Klauck as a generic lower
bound method in this model. Our result therefore uniformly implies a strong
direct product theorem for all relations for which an (asymptotically) optimal
lower bound can be provided using the smooth rectangle bound, for example Inner
Product, Greater-Than, Set-Disjointness, Gap-Hamming Distance etc. Our result
also implies near optimal direct product results for several important
functions and relations used to show exponential separations between classical
and quantum communication complexity, for which near optimal lower bounds are
provided using the rectangle bound, for example by Raz [1999], Gavinsky [2008]
and Klartag and Regev [2011]. In fact we are not aware of any relation for
which it is known that the smooth rectangle bound does not provide an optimal
lower bound. This lower bound subsumes many of the other lower bound methods,
for example the rectangle bound (a.k.a the corruption bound), the smooth
discrepancy bound (a.k.a the \gamma_2 bound) which in turn subsumes the
discrepancy bound, the subdistribution bound and the conditional min-entropy
bound.
  We show our result using information theoretic arguments. A key tool we use
is a sampling protocol due to Braverman [2012], in fact a modification of it
used by Kerenidis, Laplante, Lerays, Roland and Xiao [2012].
