A Banach space is \emph{polyhedral} if every finite dimensional subspace is isometrically isomorphic to a subspace of some finite-dimensional $\ell_\infty^n$. Banach spaces admitting a polyhedral renorming are called \emph{isomorphically polyhedral}. Many times, and in many places \cite{troy,castpapi}, has the author asked whether to be isomorphically polyhedral is a 3-space property: which means whether it is true that whenever a subspace $Y$ of a Banach space $Z$ as well as the corresponding quotient $X=Z/Y$ are isomorphically polyhedral then $Z$ has to be isomorphically polyhedral. Equivalently, if every twisted sum of two isomorphically polyhedral spaces is isomorphically polyhedral. In this talk we plan to pinpoint the difficulties (we can see) to solve this problem and present the available relevant partial results.
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