Covering $$B_X$$ by finitely many convex sets

Abstract Given a finite covering by closed convex sets of $$B_X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>X</mml:mi> </mml:msub> </mml:math> , the unit ball of an infinite-dimensional Banach space, we investigate whether there is a set of the covering that contains balls of radius close to 1 and (a) arbitrarily high finite dimension or (b) infinite dimension. In case (a) the answer is affirmative, but for the case (b) we just get radius close to 1/2 and finite codimension under much more restrictive hypotheses.