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.

中文翻译：

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反链的投影不等式

令 n 为 n ≥ 2 的整数。如果集合 A ⊆ ℝ n 不包含两个不同的元素 x = ( x 1 , …, xn ) 和 y = ( y 1 , ..., yn ) 满足 xi ≤ yi (resp. xi < yi ) 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 维至多为 n − 1 并且 A 的 ( n − 1) 维 Hausdorff 测度至多为 n ，其中是最好的边界。这个结果是作为以下投影不等式的推论推导出来的，这可能是独立的兴趣：（弱）反链 A ⊆ [0, 1] n 的 ( n -1) 维豪斯多夫测度不能超过( n − 1) 维豪斯多夫测量 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。