  This paper provides a multivariate extension of Bertoin's pathwise
construction of a L\'evy process conditioned to stay positive/negative. Thus
obtained processes conditioned to stay in half-spaces are closely related to
the original process on a compact time interval seen from its directional
extremal points. In the case of a correlated Brownian motion the law of the
conditioned process is obtained by a linear transformation of a standard
Brownian motion and an independent Bessel-3 process. Further motivation is
provided by a limit theorem corresponding to zooming in on a L\'evy process
with a Brownian part at the point of its directional infimum. Applications to
zooming in at the point furthest from the origin are envisaged.
