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On non-central sections of the simplex, the cube and the cross-polytope Koenig, Hermann


We determine the non-central hyperplane sections of the $n$-simplex of maximal volume which have a fixed large distance to the centroid - large in the sense that the distance is bigger than the distance of the centroid to the midpoint if the edges. This complements similar results of Moody, Stone, Zach and Zvavitch for the $n$-cube and of Liu and Tkocz for the $n$-cross-polytope. We also show that parallels to the extremal hyperplanes for the $n$-simplex, the $n$-cube and the $n$-cross-polytope still provide at least local maxima for smaller distances, in a specified distance range and for sufficiently large dimensions (e.g. $n \ge 10$). Moreover, we find the maximal perimeters of non-central hyperplane sections of these bodies with large distances to the center. By perimeter we mean the $(n-2)$-dimensional intersection of the hyperplane with the boundary of the convex body.

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