Intersection properties of balls in Banach spaces

We introduce a weaker notion of central subspace called almost central subspace,and we study banach spaces that belong to the class (gc),introduced by veselý (1997). in particular,we prove that if y is an almost central subspace of a banach space x such that y is in the class (gc),then y is a central subspace of x . we also prove that if y is a semi m -ideal in a banach space x such that y is an almost central subspace of x,then y is an m -ideal in x. certain stability results for quotient spaces,injective tensor product spaces,and polyhedral direct sums of banach spaces are also derived. © 2013 c. r. jayanarayanan.