Special embeddings of finite-dimensional compacta in Euclidean spaces

If $g$ is a map from a space $X$ into $mathbb R^m$ and $z otin g(X)$, let $P_{2,1,m}(g,z)$ be the set of all lines $Pi^1subsetmathbb R^m$ containing $z$ such that $|g^{-1}(Pi^1)|geq 2$. We prove that for any $n$-dimensional metric compactum $X$ the functions $gcolon Xtomathbb R^m$, where $mgeq 2n+1$, with $dim P_{2,1,m}(g,z)leq 0$ for all $z otin g(X)$ form a dense $G_delta$-subset of the function space $C(X,mathbb R^m)$. A parametric version of the above theorem is also provided.