This is one of the few problems from the Scottish book that are still open. In modern terminology, the problem is:
Let and be Banach spaces, and let be a bijective map with the following property: For every there exists such that for the sphere , the restriction is isometric. Does it follow that is isometric?
It is noted in the Scottish Book that the answer is „yes“ whenever is continuous, which is automatic if is finite-dimensional, or if has the property that for any two elements satisfying and there exists such that .