  In this paper we study the insulated conductivity problem involving two
adjacent convex insulators embedded in a bounded domain. It is known that the
gradient of solutions may blow up as the distance between two inclusions tends
to zero. For general convex insulators, we establish a pointwise upper bound
and a lower bound of the gradient with optimal blow up rates, which are
associated with the first nonzero eigenvalue of an elliptic operator determined
by the geometry of insulators. This extends the previous result for ball
insulators in \cite{DLY}.
