  We show that a constant factor approximation of the shortest and closest
lattice vector problem in any norm can be computed in time $2^{0.802\, n}$.
This contrasts the corresponding $2^n$ time, (gap)-SETH based lower bounds for
these problems that even apply for small constant approximation. For both
problems, $\mathrm{SVP}$ and $\mathrm{CVP}$, we reduce to the case of the
Euclidean norm. A key technical ingredient in that reduction is a twist of
Milman's construction of an $M$-ellipsoid which approximates any symmetric
convex body $K$ with an ellipsoid $\mathcal{E}$ so that $2^{\varepsilon n}$
translates of a constant scaling of $\mathcal{E}$ can cover $K$ and vice versa.
