Scottish Book Problem 155

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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