Some observations on the numerical index and the polynomial numerical index (Q2787650)

The paper under review is devoted to present some facts about the numerical index and the polynomial numerical index of Banach spaces. More precisely, it is proved that a Banach space with the alternative Daugavet property whose unit ball has small combinations of slices is lush (a property of geometric nature which implies numerical index one). In the polynomial setting, the author proves a few results concerning renormings which are similar to previously known ones in the linear case. These results allow to deduce that the set of values under equivalent renorming of the polynomial numerical index of order \(k\) of a Banach space is an interval, which is shown to be non-trivial in the real case for \(k=2\) by means of an inequality for the second-order polynomial numerical index in terms of the constant of the \(\beta\) property for Banach spaces.