立方体切片和板的临界矩（及其他多面体范数）
Critical moments of slices and slabs of the cube (and other polyhedral norms)
AI总结：
提出统一代数-组合框架，计算多面体范数球切片和板的体积与高阶矩的显式分段有理参数公式，并应用于低维立方体，发现新有理函数族和临界点。
摘要：
在本文中，我们提出了一个统一的代数-组合框架，用于计算多面体范数球的切片和板的体积和高阶矩的显式、分段有理且组合索引的参数公式。我们的主要方法建立在先前关于多面体所有切片的参数空间的组合分解的工作之上。我们将此框架扩展到板，并发现了固定维度下的多项式时间算法。我们还展示了获得任意阶矩的计算方法，适用于任何多面体范数球的所有切片或板，以及一个用于分析其临界点的代数框架。此外，我们对$d$维单位立方体进行了实验研究。我们的分析恢复并重新解释了König和Koldobsky首次获得的二维和三维立方体的板和切片的已知体积公式。此外，我们的方法识别了一个新的由十四个有理函数组成的完整族，给出了四维立方体的切片和板的体积。我们进一步计算了二维和三维中切片和板的显式高阶矩，并导出了二维立方体切片任意阶矩的显式公式，描述了它们的临界点。对于四维立方体，我们进一步利用这些公式来识别切片和板体积的候选全局最大值和最小值；通过精确符号计算验证这些候选点为临界点，并通过数值检查确定为全局极值。
英文摘要：
In this article, we present a unified algebraic-combinatorial framework for computing explicit, piecewise rational, and combinatorially indexed parametric formulas for volumes and higher moments of slices and slabs of polyhedral norm balls. Our main method builds on prior work concerning a combinatorial decomposition of the parameter space of all slices of a polytope. We extend this framework to slabs, and find a polynomial-time algorithm in fixed dimension. We also exhibit computational methods to obtain moments of arbitrary order for all slices or slabs of any polyhedral norm ball, and an algebraic framework for analyzing their critical points. In addition, we present an experimental study of the $d$-dimensional unit cube. Our analysis recovers and reinterprets the known volume formulas for slabs and slices of the two- and three-dimensional cubes, first obtained by König and Koldobsky. Moreover, our method identifies a new complete family of fourteen rational functions giving the volumes of slices and slabs of the four-dimensional cube. We further compute explicit higher moments of slices and slabs in dimensions two and three, and derive explicit formulas for moments of arbitrary order for slices of the two-dimensional cube, describing their critical points. For the four-dimensional cube, we further use these formulas to identify candidate global maxima and minima for slice and slab volumes; the candidates are verified to be critical points by exact symbolic computation and are checked numerically to give the global extrema.
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。
