On mappings ``lossening'' convexity (Q1921849)

The article deals with a further generalization of assertion characterizing affine maps in \(\mathbb{R}^n\) in terms of preserving convexity. The main result can be formulated as follows. For a given real number \(a>0\), a set \(M\subset \mathbb{R}^n\) is called \(a\)-almost convex if there is a convex set \(B\subset \mathbb{R}^n\) and a measurable set \(C\subset \mathbb{R}^n\) of Lebesgue measure \(m(C)\leq a\), such that \(M=B \cup C\). Theorem. Let \(M_0 \subset \mathbb{R}^n\), \(n\geq 2\), be a convex body distinct from \(\mathbb{R}^n\), and \(a>0\) be a real number. If \(f:\mathbb{R}^n \to \mathbb{R}^n\) is a bijective map such that the image \(f(M)\) is \(a\)-almost convex for any convex body \(M\subset \mathbb{R}^n\) similar to \(M_0\), then \(f\) is affine.