  We revisit the damped wave equation on two-dimensional torus where the damped
region does not satisfy the geometric control condition. We show that if the
damping vanishes as a H\"older function $|x|^{\beta}$, and in addition, the
boundary of the damped region is strictly convex, the wave is stable at rate
$t^{-1+\frac{2}{2\beta+7}}$, which is better than the known optimal decay rate
$t^{-1+\frac{1}{\beta+3}}$ for strip-shaped dampings of the same H\"older
regularity. Moreover, we show by example that the decay rate is optimal. This
illustrates the fact that the energy decay rate depends not only on the order
of vanishing of the damping, but also on the shape of the damped region. The
main ingredient of the proof is the averaging method (normal form reduction)
developed by Hitrick and Sj\"ostrand (\cite{Hi1}\cite{Sj}).
