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Tarski's Planks problem, solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires "planks" (parallel strips) of total width $\ge d$ in order to completely cover a disk of diameter $d$. The last paragraph of Bang's short paper poses an open problem:

          Bang
          (Snapshot from Bang's paper.)
Does anyone know if this problem has been solved? I am especially interested in $\mathbb{R}^2$.

Bang, Thøger (1951), "A solution of the 'plank problem'", Proc. Amer. Math. Soc. 2 (6): 990–993, doi:10.2307/2031721, JSTOR 2031721, MR 0046672.

Martin Sleziak
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Joseph O'Rourke
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    I don't think it solves the problem, but it is discussed in Bezdek's paper, Tarski's plank problem revisited, available at http://arxiv.org/pdf/0903.4637.pdf – Gerry Myerson Sep 21 '15 at 23:41
  • @GerryMyerson: Great resource, thanks! "Recall that Ball ([2]) proved that such sum [of relative widths of the body] should exceed $1$ in the case of centrally symmetric body $K$, while the general case is still open." – Joseph O'Rourke Sep 21 '15 at 23:51
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    It is open, as I know. Solved in partial cases, in particular for centrally symmetric bodies (K. Ball). – Fedor Petrov Sep 22 '15 at 00:09

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