The Mazur-Ulam property for two-dimensional somewhere-flat spaces

In this paper, we show that if X is a two-dimensional real normed space such that its unit sphere contains a line segment with the distance between its endpoints being greater than 1, then X has the Mazur-Ulam property. That is, every isometry from the unit sphere of X onto the unit sphere of any normed space Y can be extended to a linear isometry from X onto Y.