Maps which almost preserve convexity (Q1118166)

It is shown that if \(f: L\to L'\) is an injective mapping between two real-linear spaces L and \(L'\) (\(\dim L\geq 2\) and \(\dim L'\geq 1)\) such that f(M) is almost convex for all M from a sufficiently large family \({\mathcal M}\) of convex sets, then f is affine, i.e. there exists a linear injective \(\phi: L\to L'\) and a translation \(t: L'\to L'\) such that \(f=t\circ \phi\). (Here, almost convexity may be in Lebesgue-, resp. Baire-sense: \(M'=f(M)\subset L'\) is Lebesgue-, resp. Baire-convex iff for every \(x,y\in M'\) the set \([x,y]\setminus M'\) is of zero-measure, resp. is a set of first category. The family \({\mathcal M}\) must be sufficiently large that for every line \(\ell \subset L\) and point \(X\in \ell\) there exists an open segment \(I(X,\ell): X\in I(X,\ell)\subset \ell\), such that every open segment \(I\subset I(X,\ell)\) is a countable intersection of elements from \({\mathcal M}.)\)