We present algorithms for the $(1+\epsilon)$-approximate version of the
closest
vector problem for certain norms. The currently fastest algorithm
(Dadush and
Kun 2016) for general norms has running time $2^{O(n)} (1/\epsilon)^n$. We
improve this substantially for convex bodies whose modulus of smoothness, a
quantity expressing how well tangent hyperplanes approximate the
boundary, is
well bounded. This is the case for unit balls of $\ell_p$ spaces. Joint
work
with Moritz Venzin.