Abstract
A generalization is given of the work of J. Bokowski [1] referring to the product of the volumes of the two parts into which a convex body is divided by a plane. The proof uses formulas of Integral Geometry and a conjecture of L. A. Santaló [2], and holds for the two parts determined by any (n−1)-dimensional surface in the euclideann-space and for dimensionsn=2, 3 in the hyperbolic space.
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References
Bokowski J.,Ungleichungen für den Inhalt von Trennflächen Arch. Math. (Basel)34 (1980) pp. 84–89.
Santaló L.A.,An inequality between the parts into which a convex body is divided by a plane section, Rendiconti Del Circolo Matematico Di Palermo, Serie II, Tomo XXXII (1983) pp. 124–130.
Santaló L.A.,Integral geometry and geometric probability, Encyclopedia of Math. and Applications, Addison-Wesley, Reading, 1976.
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Gysin, L.M. Inequalities for the product of the volumes of a partition determined in a convex body by a surface. Rend. Circ. Mat. Palermo 35, 420–428 (1986). https://doi.org/10.1007/BF02843909
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DOI: https://doi.org/10.1007/BF02843909