Banach Spaces with an Infinite Number of Smooth Faces in their Unit Ball

We study Banach spaces having smooth faces in their unit ball. In particular, we show that if the unit ball of a finite dimensional Banach space has an infinite number of smooth faces then their interiors relative to the unit sphere approach the empty set in a certain way. We also show that this situation does not hold in infinite dimensions since we prove that every infinite dimensional Banach space can be equivalently renormed to have infinitely many smooth faces with interior relative to the unit sphere of the same "size". This fact characterizes having infinite algebraic dimension.