  On a large class of Riemannian manifolds with boundary, some dimension-free
Harnack inequalities for the Neumann semigroup is proved to be equivalent to
the convexity of the boundary and a curvature condition. In particular, for
$p_t(x,y)$ the Neumann heat kernel w.r.t. a volume type measure $\mu$ and for
$K$ a constant, the curvature condition $\Ric-\nn Z\ge K$ together with the
convexity of the boundary is equivalent to the heat kernel entropy inequality
$$\int_M p_t(x,z)\log \ff{p_t(x,z)}{p_t(y,z)} \mu(\d z)\le
\ff{K\rr(x,y)^2}{2(\e^{2Kt}-1)}, t>0, x,y\in M,$$ where $\rr$ is the Riemannian
distance. The main result is partly extended to manifolds with non-convex
boundary and applied to derive the HWI inequality.
