An investigation is carried out of the compact convex sets X in an infinite-dimensional separable Hilbert space 𝔼, for which the metric antiprojection $q_X(e)$ from e to X has fixed cardinality n+1 ($n ⊆ ℕ$ arbitrary) for every e in a dense subset of 𝔼. A similar study is performed in the case of the metric projection $p_X(e)$ from e to X where X is a compact subset of 𝔼.
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