Abstract
We generalize an old problem and its partial solution of Pólya (Pólya in Elem. Math. 13, 40–41 (1958)) to the n-dimensional setting. Given a plane domain Ω⊂ℝ2, Pólya asked in 1958 for the shortest bisector of Ω, that is for the shortest line segment l(Ω) which divides Ω into two subsets of equal area. He claimed that among all centrosymmetric domains of given area l(Ω) becomes longest for a disk. His proof, however, does not seem to be valid for domains that are not starshaped with respect to the center of Ω. In the present note we provide two proofs that it suffices to restrict attention to starshaped sets. Moreover we state and prove a related inequality in ℝn. Given the volume of a measurable set Ω with finite Lebesgue measure, only a ball centered at zero maximizes the length of the shortest line segments running through the origin. In this sense the ball has the longest shortest piercing.
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Acknowledgements
This research is financially supported by the Alexander von Humboldt Foundation (AvH). The second author is very grateful to AvH for the great opportunity to visit the Mathematical Institute of Köln University and to Professor B. Kawohl for his cordial hospitality during this visit.
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Kawohl, B., Kurta, V.V. (2012). The Longest Shortest Piercing. In: Bandle, C., Gilányi, A., Losonczi, L., Plum, M. (eds) Inequalities and Applications 2010. International Series of Numerical Mathematics, vol 161. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0249-9_8
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DOI: https://doi.org/10.1007/978-3-0348-0249-9_8
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