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Large antipodal families

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A family {A i | i ∈ I} of sets in ℝ d is antipodal if for any distinct i, j ∈ I and any p ∈ A i , q ∈ A j , there is a linear functional ϕ:ℝ d → ℝ such that ϕ(p) ≠ ϕ(q) and ϕ(p) ≤ ϕ(r) ≤ ϕ(q) for all r ∈ ∪ i∈I A i . We study the existence of antipodal families of large finite or infinite sets in ℝ3.

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