  Given a symmetric convex body $C$ and $n$ hyperplanes in an Euclidean space,
there is a translate of a multiple of $C$, at least ${1\over n+1}$ times as
large, inside $C$, whose interior does not meet any of the hyperplanes. The
result generalizes Bang's solution of the plank problem of Tarski and has
applications to Diophantine approximation.
