Title: On certain geometric properties in Banach spaces of vector-valued functions,
Abstract: We consider a certain type of geometric properties of Banach spaces, which
includes for instance octahedrality, almost squareness, lushness and the
Daugavet property. For this type of properties, we obtain a general reduction
theorem, which, roughly speaking, states the following: if the property in
question is stable under certain finite absolute sums (for example finite
$\ell^p$-sums), then it is also stable under the formation of corresponding
Köthe-Bochner spaces (for example $L^p$-Bochner spaces). From this general
theorem, we obtain as corollaries a number of new results as well as some
alternative proofs of already known results concerning octahedral and almost
square spaces and their relatives, diameter-two-properties, lush spaces and
other classes.