Subspaces of Banach spaces with big slices

Abstract

We study when diameter $2$ properties can be inherited by subspaces. We obtain that the slice diameter $2$ property (resp., the diameter $2$ property, strong diameter $2$ property) passes from a Banach space $X$ to a subspace $Y$ whenever $X/Y$ is finite-dimensional and $Y$ is complemented by a norm $1$ projection (resp., the quotient $X/Y$ is finite-dimensional and strongly regular). Also, we study the same problem for the dual properties of diameter $2$ properties, such as having octahedral, weakly octahedral, or $2$-rough norm.