  In this paper, we obtain estimates on the quantitative strata of the critical
set of non-trivial harmonic functions $u$ which vanish continuously on $V
\subset \partial \Omega$, a relatively open subset of the boundary of a convex
domain $\Omega \subset \mathbb{R}^n$. In particular, these estimates improve
dimensional estimates on $\{|\nabla u| =0\}$ both in $V \subset \partial
\Omega$ and as it \textit{approaches} $V \cap \overline{\Omega}.$ These
estimates are not obtainable by naively combining interior and boundary
estimates and represent a significant improvement upon existing results for
boundary analytic continuation in the convex case.
