Any isometry between the spheres of absolutely smooth 2-dimensional Banach spaces is linear

A 2-dimensional Banach space Xis called absolutely smoothif its unit sphere is the image of the real line under a differentiable function r : R -> SX whose derivative is locally absolutely continuous and has parallel to r' (s)parallel to = 1for all s is an element of R. We prove that any isometry f: S-X -> S-Y between the unit spheres of absolutely smooth Banach spaces X, Yextends to a linear isometry (f) over bar: X -> Yof the Banach spaces X, Y. This answers the famous Tingley's problem in the class of absolutely smooth 2dimensional Banach spaces. (C) 2021 Elsevier Inc. All rights reserved.