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

Abstract

A 2-dimensional Banach space X is called absolutely smooth if 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 ‖r′(s)‖=1 for all s∈R. We prove that any isometry f:SX→SY between the unit spheres of absolutely smooth Banach spaces X,Y extends to a linear isometry f¯:X→Y of the Banach spaces X,Y. This answers the famous Tingley's problem in the class of absolutely smooth 2-dimensional Banach spaces.