The Mazur-Ulam theorem (1932) states that any isometry of a normed linear space is affine. See Nica (Expo. Math. 30 (2012), 397-398; arXiv:1306.2380) for a very elegant proof.
Question: Let $M$ be a closed convex set with empty interior in a normed linear space and $T:M\to M$ be an isometry of $M$ onto itself. Does it follow that $T$ is affine (maps linear segments to linear segments)?
This question naturally arises in connection with the work of Bader, Furman, Gelander, Monod, see p. 88 of their paper `Property (T) and rigidity for actions on Banach spaces' (Acta Math. 198 (2007), 57-105; arXiv:math/0506361).
Related information: 1. The question has already been answered in the positive by Mankiewicz (On extension of isometries in normed linear spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 20 (1972), 367-371) for closed convex sets having nonempty interior.
- There is an active related work on the problem suggested by Tingley (Geometriae Dedicata, 1987, p.371): Suppose that $f: S_X\to S_Y$is an (onto) isometry between spheres of normed spaces. Is $f$ necessarily the restriction to $S_X$ of a linear, or affine, transformation? But in this work a starting point is an isometry of a non-convex set.