A in the unit @math -cube is a set @math such that for every @math and @math in @math we either have @math for all @math , or @math for all @math . We consider subsets, @math , of the unit @math -cube @math that satisfy [ card (A C) k, , for all chains , C [0,1]^n , , ] where @math is a fixed positive integer. We refer to such a set @math as a @math -antichain. We show that the @math -dimensional Hausdorff measure of a @math -antichain in @math is at most @math and that the bound is asymptotically sharp. Moreover, we conjecture that there exist @math -antichains in @math whose @math -dimensional Hausdorff measure equals @math and we verify the validity of this conjecture when @math .
When @math this conjecture is clearly true, and when @math it is observed in @cite_0 that the validity of Conjecture is an immediate consequence of the following, well-known, result. Recall that a singular function @math is a strictly decreasing function whose derivative equals zero almost everywhere.
@cite_0
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A set @math is called an antichain (resp. antichain) if it does not contain two distinct elements @math and @math satisfying @math (resp. @math ) for all @math . We show that the Hausdorff dimension of a weak antichain @math in the @math -dimensional unit cube @math is at most @math and that the @math -dimensional Hausdorff measure of @math is at most @math , 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 @math -dimensional Hausdorff measure of a (weak) antichain @math cannot exceed the sum of the @math -dimensional Hausdorff measures of the @math orthogonal projections of @math onto the facets of the unit @math -cube containing the origin. For the proof of this result we establish a discrete variant of the projection inequality applicable to weak antichains in @math and combine it with ideas from geometric measure theory.