Recognition and reconstruction of sets in \(\ell^2\) via their projections (Q6546623)

Let \(k\in \mathbb{N}\) and let \(\mathcal{P}\) be a subset of all \(k\)-dimensional linear subspaces \(\mathcal{G}_k\) of \(\ell^{2}\) with the natural topology. The subsets \(B\) and \(C\) of \(\ell^{2}\) are called \(\mathcal{P}\)-imitations of each other if \(B+P=C+P\) for every \(P\in \mathcal{P} \). The author points out that, for \(\mathcal{P}\) a somewhere dense \(\mathcal{G}_\delta\)-set in \(\mathcal{G}_k\), there are certain non-trivial sets in \(\ell^{2}\) such that each of them has only one \(\mathcal{P}\)-imitation, namely, itself and further every such set can be reconstructed as the intersection of the preimages of its projections under \(\mathcal{P}\). In addition, the author discusses some important properties of \(\sigma\)-compact sets that are of independent interest.