单形、交叉多面体和立方体的非中心截面
Non-central sections of the simplex, the cross-polytope and the cube
摘要：
如果正则$n$ -单形的超平面到形心的距离相当大，即大于形心到边中点的距离，则确定正则$n$ -单形的最大超平面截面. Moody、Stone、Zach和Zvavitch以及Liu和Tkocz对n-立方体和l_1-球也得到了类似的结果。在这三种情况下的极大超平面垂直于从质心到顶点的向量。对于较小的距离-在一个明确定义的范围内-我们表明，这些超平面部分至少是局部最大的。我们还确定了单形、交叉多面体和立方体与凸体边界具有最大周长即最大体积相交的超平面截面。
英文摘要：
We determine the maximal hyperplane sections of the regular $n$-simplex, if the distance of the hyperplane to the centroid is fairly large, i.e. larger than the distance of the centroid to the midpoint of edges. Similar results for the n-cube and the l_1-ball were obtained by Moody, Stone, Zach and Zvavitch and by Liu and Tkocz. The maximal hyperplanes in these three cases are perpendicular to the vectors from the centroid to the vertices. For smaller distances -- in a well-defined range -- we show that these hyperplane sections are at least locally maximal. We also determine the hyperplane sections of the simplex, the cross-polytope and the cube which have maximal perimeter, i.e. maximal volume intersection with the boundary of the convex body.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
