Nonrough norms in spaces with small diameter

Abstract: In this work, we study the non rough norms in Banach spaces with small diameter properties, namely the Ball Dentable Property (BDP), the Ball Huskable Property (BHP) and the Ball Small Combination of Slice Property (BSCSP). We introduce two more notions of non rough norms, namely the weakly average non rough norms and average non rough norms in Banach spaces. We prove the duality between these three versions of non rough norms and the small diameter properties in Banach spaces. We also prove that each of the three non rough norms is a three space property under certain assumptions.