  Electrical Impedance Imaging would suffer a serious obstruction if for two
different conductivities the potential and current measured at the boundary
were the same. The Calder\'on's problem is to decide whether the conductivity
is indeed uniquely determined by the data at the boundary. In $\mathbb{R}^d$,
for $d=5,6$, we show that uniqueness holds when the conductivity is in
$W^{1+\frac{d-5}{2p}+, p}(\Omega)$, for $d\le p <\infty$. This improves on
recent results of Haberman, and of Ham, Kwon and Lee. The main novelty of the
proof is an extension of Tao's bilinear Theorem.
