  The aim of this note is to study some geometrical properties like diameter
two properties, octahedrality and almost squareness in the setting of
(symmetric) tensor product spaces. In particular, we show that the injective
tensor product of two octahedral Banach spaces is always octahedral, the
injective tensor product of an almost square Banach space with any Banach space
is almost square, and the injective symmetric tensor product of an octahedral
Banach space is octahedral.
