Abstract
First, we give a proof of the long-standing affine plank conjecture of Bang for successive hyperplane cuts and then for inductive partitions. Second, we prove a lower estimate for the sum of the cross-sectional volumes of cylinders covering a convex body in Euclidean d-space. Then we prove a Kadets–Ohmann-type theorem in spherical d-space for coverings of balls by convex bodies via volume maximizing lunes. Finally, we give estimates for partial coverings of balls by planks in Euclidean d-space.
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Bezdek, K. (2013). Proofs on Coverings by Cylinders. In: Lectures on Sphere Arrangements – the Discrete Geometric Side. Fields Institute Monographs, vol 32. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8118-8_8
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DOI: https://doi.org/10.1007/978-1-4614-8118-8_8
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