Certain 2-stable embeddings (Q1371927)

In the 1930's Chogoshvili made the assertion that for each \(k\)-dimensional compactum \(X\subset \mathbb{R}^n\), there exists an \((n-k)\)-dimensional plane \(P\subset\mathbb{R}^n\) such that for some \(\varepsilon>0\), every map \(f:X\to \mathbb{R}^n\) with \(|x-f(x) |< \varepsilon\) for all \(x\in X\), has the property that \(f(X) \cap P\neq \emptyset\). A gap in his proof was found in the 1980's and recently, \textit{A. N. Dranishnikov} constructed a counterexample [Proc. Am. Math. Soc. 125, No. 7, 2155-2160 (1997; Zbl 0879.55001)]. The authors show that, on the other hand, Chogoshvili's claim is valid for each 2-dimensional \(LC^1\)-compactum, and hence each 2-dimensional disk. They also establish a local path-connectification of Dranishnikov's counterexample.