  $\Xi^-$ escaping and scattering probabilities in the $(K^-,K^+)$ reaction
with nuclear targets is investigated on the basis of the eikonal approximation.
Calculations are performed using the $S=-2$ sector of the Nijmegen model-D
interaction, whose hard core radii are treated phenomenologically. %The $\Xi^-
N$, $\Xi^- p$ and $\Xi^- n$ elastic cross sections are The $\Xi^- N$ elastic
cross sections are derived from the $^9$Be$(K^-,K^+)$ data of the BNL-E906
experiment, which is useful to determine the hard-core radii in $P$-states. The
$\Xi$ single-particle potential energy is investigated with the G-matrix
calculation in nuclear matter, which is sensitive to the hard-core radii in
$S$-states. The important constraint for the $\Xi N$ interaction is obtained by
combining the two analyses.
