论文标题
当地的超立方体部分
Local extrema for hypercube sections
论文作者
论文摘要
考虑从HyperCube $ [0,1]^d $的固定距离$ t $的固定距离$ t $。在这些问题中，非常重视确定超平面$ h $，以便$（d-1）$ - 尺寸$ h \ cap [0,1]^d $是最大或最小的。本着维塔利·米尔曼（Vitali Milman）提出的问题的精神，当$ h $与对角线或超立方体的对角线正交时，这里考虑了相应的本地问题。事实证明，该卷在对角线上严格在所有维度上最大程度大于$ 3 $的$ t $，该范围渐近至$ \ sqrt {d}/\！\ log d $。在较低订单的子二角形下，当$ t $接近$ 0 $，而当$ t $大于$ t $时，该卷被证明是严格的本地最大程度。这依赖于对角和子对角线的局部极值的表征，该分子允许在任何固定的，相当低的尺寸中以$ t $的整个可能的范围解决问题。
Consider the hyperplanes at a fixed distance $t$ from the center of the hypercube $[0,1]^d$. Significant attention has been given to determining the hyperplanes $H$ among these such that the $(d-1)$-dimensional volume of $H\cap[0,1]^d$ is maximal or minimal. In the spirit of a question by Vitali Milman, the corresponding local problem is considered here when $H$ is orthogonal to a diagonal or a sub-diagonal of the hypercube. It is proven in particular that this volume is strictly locally maximal at the diagonals in all dimensions greater than $3$ within a range for $t$ that is asymptotic to $\sqrt{d}/\!\log d$. At lower order sub-diagonals, this volume is shown to be strictly locally maximal when $t$ is close to $0$ and not locally extremal when $t$ is large. This relies on a characterisation of local extremality at the diagonals and sub-diagonals that allows to solve the problem over the whole possible range for $t$ in any fixed, reasonably low dimension.
