Recall that a Banach space \(X\) has the Daugavet property if every rank-one operator \(T:X\longrightarrow X\) satisfies the so-called Daugavet equation \[\Vert \mathrm{Id}-T\Vert=1+\Vert T\Vert.\] One of the highlights of the Daugavet property is the following geometric characterisation: a Banach space \(X\) has the Daugavet property if, and only if, for every \(x\in S_X\), every slice \(S\) of \(B_X\) and every \(\varepsilon>0\) there is an element \(y\in S\) so that \(\Vert x-y\Vert>2-\varepsilon\).
The connection between the Daugavet equation and the geometry of slices of a Banach space has been shown to be much deeper. For instance, in [Y. Ivakhno and V. Kadets, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 645, No. 54, 30–35 (2004; Zbl 1071.46015)], the authors considered the following weaker property: a Banach space \(X\) is said to be a space with bad projections if \(\Vert \mathrm{Id}-P\Vert\geq 2\) for every rank-one projection \(P:X\longrightarrow X\). In that paper, it is proved that a Banach space \(X\) is a space with bad projections if, and only if, given any slice \(S\) of \(B_X\), any \(x\in S\cap S_X\) and any \(\varepsilon>0\), there exists \(y\in S\) so that \(\Vert x-y\Vert\geq 2-\varepsilon\) (i.e., every slice of \(B_X\) has diameter two and every point of \(S\cap S_X\) is diametral).
The previous characterisation motivated J. Becerra Guerrero et al. [J. Convex Anal. 25, No. 3, 817–840 (2018; Zbl 1403.46008)] to define the following strenghtening of the notion of spaces with bad projections:

1.

\(X\) is said to have the diametral diameter two property (DD2P) if, given any non-empty relatively weakly open subset \(W\) of \(B_X\), any \(x\in W\cap S_X\) and any \(\varepsilon>0\), there exists \(y\in W\) so that \(\Vert x-y\Vert>2-\varepsilon\).

2.

\(X\) has the diametral strong diameter two property (DSD2P) if, for every convex combination of non-empty relatively weakly open subsets \(C\) of \(B_X\), every \(x\in C\) and every \(\varepsilon>0\), there exists \(y\in C\) so that \(\Vert x-y\Vert>1+\Vert x\Vert-\varepsilon\).

In Example 3.3 of [loc. cit.] it is proved that if \(X\) has the Daugavet property, then \(X\) has the DSD2P, and it remained open whether the converse is true (even since its first preprint version in 2015).
The aim of the paper under review is to prove that this converse holds true, hence the DSD2P implies the Daugavet property, solving the above mentioned open question. The proof, far from being technical, uses nothing but homogeneization arguments of \(\ell_1\)-like inequalities, but they are used in a brilliant way together with a smart and sharp work with slices. The proof is quite beautiful and readable.