Projection inequalities for antichains
Projection inequalities for antichains
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反链的投影不等式
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Christian Reiher
中科院分区：
文献类型：
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作者：
K. Engel;T. Mitsis;C. Pelekis;Christian Reiher
设n为整数，n ≥ 2。一个集合A的反链被称为反链（antichain）。弱反链），如果它不包含两个不同的元素x =（x1，...，xn）和y =（y1，...，yn），满足xi ≤ yi（分别为x i < y i）对于所有i ∈ {1，.，n }。我们证明了n维单位立方体[0，1] n中弱反链A的Hausdorff维数至多为n-1，A的（n-1）维Hausdorff测度至多为n，这是最佳可能界.这个结果是由下列投影不等式导出的，它可能是独立的：（弱）反链A <$[0，1] n的（n −1）维Hausdorff测度不能超过A在包含原点的单位n -立方体的小平面上的n个正交投影的（n − 1）维Hausdorff测度之和。为了证明这一结果，我们建立了一个适用于弱反链的投影不等式的离散变形，并将其与几何测度论的思想联合收割机结合起来。
Let n be an integer with n ≥ 2. A set A ⊆ ℝ n is called an antichain (resp. weak antichain) if it does not contain two distinct elements x = ( x 1 , …, x n ) and y = ( y 1 , …, y n ) satisfying x i ≤ y i (resp. x i < y i ) for all i ∈ {1, …, n }. We show that the Hausdorff dimension of a weak antichain A in the n -dimensional unit cube [0, 1] n is at most n − 1 and that the ( n − 1)-dimensional Hausdorff measure of A is at most n , which are the best possible bounds. This result is derived as a corollary of the following projection inequality, which may be of independent interest: The ( n −1)- dimensional Hausdorff measure of a (weak) antichain A ⊆ [0, 1] n cannot exceed the sum of the ( n − 1)-dimensional Hausdorff measures of the n orthogonal projections of A onto the facets of the unit n -cube containing the origin. For the proof of this result we establish a discrete variant of the projection inequality applicable to weak antichains in ℤ n and combine it with ideas from geometric measure theory.
关于连续立方体中反链的注释
DOI：
10.17863/cam.101774
发表时间：
2020
期刊：
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影响因子：
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作者：
Janzer B
通讯作者：
Janzer B