Abstract
We prove that the largest convex shape that can be placed inside a given convex shape \(Q \subset \mathbb {R}^{d}\) in any desired orientation is the largest inscribed ball of Q. The statement is true both when “largest” means “largest volume” and when it means “largest surface area”. The ball is the unique solution, except when maximizing the perimeter in the two-dimensional case.
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This work was initiated at the 21st Korean Workshop on Computational Geometry, held in Rogla, Slovenia, in June 2018. We thank all workshop participants for their helpful comments.
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Cabello, S., Cheong, O. & Dobbins, M.G. The inverse Kakeya problem. Period Math Hung 84, 70–75 (2022). https://doi.org/10.1007/s10998-021-00392-z
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DOI: https://doi.org/10.1007/s10998-021-00392-z