On Daugavet indices of thickness

Abstract

Inspired by R. Whitley's thickness index the last named author recently introduced the Daugavet index of thickness of Banach spaces. We continue the investigation of the behavior of this index and also consider two new versions of the Daugavet index of thickness, which helps us solve an open problem which connects the Daugavet indices with the Daugavet equation. Moreover, we will improve formerly known estimates of the behavior of Daugavet index on direct sums of Banach spaces by establishing sharp bounds. As a consequence of our results we prove that, for every 0<δ<2, there exists a Banach space where the infimum of the diameter of convex combinations of slices of the unit ball is exactly δ, solving an open question from the literature. Finally, we prove that an open question posed by Ivakhno in 2006 about the relation between the radius and diameter of slices has a negative answer.