Projection inequalities for antichains

Abstract

Let n be an integer with n ≥ 2. A set A ⊆ ℝⁿ is called an antichain (resp. weak antichain) if it does not contain two distinct elements x = (x₁, …, x_n) and y = (y₁, …, 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 ℤⁿ and combine it with ideas from geometric measure theory.