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
.