Diameter, radius and Daugavet index of thickness of slices in Banach spaces

We construct a Banach space $X$ with the r-BSP such that the infimum of the diameter of the slices of the unit ball is $1$, which gives negative answer to a 2006 question by Y. Ivakhno in an extreme way. This example is performed by considering modifications of the classical James-tree space $J{T}_{\mathrm{\infty }}$ constructed on a tree with infinitely many branching points $T_{\mathrm{\infty }}$. Moreover we prove that every Banach space with the Daugavet property admits, for every $\epsilon >0$, an equivalent renorming for which its Daugavet index of thickness is bigger than $2-\epsilon$ and there are slices of the unit ball of diameter strictly smaller than $2$, which solves an open question from [7].