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Plank theorems via successive inradii

In the 1930's, Tarski introduced his plank problem at a time when the field discrete geometry was about to born. It is quite remarkable that Tarski's question and its variants continue to generate interest in the geometric as well as analytic aspects of coverings by planks in the present time as well. Besides giving a short survey on the status of the affine plank conjecture of Bang (1950) we prove some new partial results for the successive inradii of the convex bodies involved. The underlying geometric structures are successive hyperplane cuts introduced several years ago by Conway and inductive tilings introduced recently by Akopyan and Karasev.