Scottish Book Problem 155

Von | Oktober 7, 2020

This is one of the few problems from the Scottish book that are still open. In modern terminology, the problem is:

Let X and Y be Banach spaces, and let U\colon X\to Y be a bijective map with the following property: For every x_0\in X there exists \varepsilon>0 such that for the sphere S(x_0,\varepsilon) := \{ x\in X : \|x-x_0\|=\varepsilon \}, the restriction U|_{S(x_0,\varepsilon)} is isometric. Does it follow that U is isometric?

It is noted in the Scottish Book that the answer is „yes“ whenever U^{-1} is continuous, which is automatic if Y is finite-dimensional, or if Y has the property that for any two elements y_1,y_2\in Y satisfying y_2\neq 0 and \|y_1+y_2\|=\|y_1\|+\|y_2\| there exists \lambda\geq 0 such that y_1=\lambda y_2.

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