The plank problem for symmetric bodies

Abstract

Given a symmetric convex body $C$ and $n$ hyperplanes in an Euclidean space, there is a translate of a multiple of $C$, at least $\frac{1}{n+1}$ 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.

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