发表机构
Warsaw University of Technology（华沙理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文将Vaaler定理的锐利表面积推广从二维和三维扩展到所有维度，证明多胞形边界测度下界，并确认Ivanov猜想，同时提出脊骨架猜想及部分结果。
AI 中文摘要
Karasev在二维和三维中证明了Vaaler定理的多面体推广的锐利表面积对应形式[8，定理1.2]。我们移除了维度限制。更精确地说，如果一个n维凸多胞形P在其内部包含原点，并且每个余维数k∈{1，…，n}的非空真面的仿射包到原点的距离至少为√k，那么H^{n-1}(∂P)≥n2^n。因此，立方体[-1,1]^N的每个n维线性截面的边界具有至少n2^n的(n-1)维测度。这证实了Karasev记录的Grigory M. Ivanov的猜想[8，第1节]在所有维度上的成立。证明结合了Rogers-Karasev旗分解与无维度的高斯比较用于正斜体。其主要步骤是有序平方替换，它将所有面距离假设转化为单个正积分的逐点支配。我们还反驳了对所有骨架测度的朴素扩展，提出了脊骨架猜想，并建立了两个部分结果。
英文摘要
Karasev proved, in dimensions two and three, a sharp surface-area counterpart of a polyhedral extension of Vaaler's theorem [8, Theorem 1.2]. We remove the dimension restriction. More precisely, if an $n$-dimensional convex polytope $P$ contains the origin in its interior and the affine hull of every nonempty proper face of codimension $k\in\{1,\ldots,n\}$ is at distance at least $\sqrt{k}$ from the origin, then $$ \mathcal{H}^{n-1}(\partial P)\geqslant n2^n. $$ Consequently, the boundary of every $n$-dimensional linear section of the cube $[-1,1]^N$ has $(n-1)$-dimensional measure at least $n2^n$. This confirms, in all dimensions, a conjecture of Grigory M. Ivanov recorded by Karasev [8, Section 1]. The proof combines the Rogers--Karasev flag decomposition with a dimension-free Gaussian comparison for orthoschemes. Its main step is an ordered-square substitution which converts all face-distance assumptions into a pointwise domination of a single positive integral. We also disprove the naive extension to all skeletal measures, formulate a ridge-skeleton conjecture, and establish two partial results toward it.
Comments12 pages
