  Fawzi and Fawzi recently defined the sharp R\'enyi divergence, $D_\alpha^\#$,
for $\alpha \in (1, \infty)$, as an additional quantum R\'enyi divergence with
nice mathematical properties and applications in quantum channel discrimination
and quantum communication. One of their open questions was the limit ${\alpha}
\to 1$ of this divergence. By finding a new expression of the sharp divergence
in terms of a minimization of the geometric R\'enyi divergence, we show that
this limit is equal to the Belavkin-Staszewski relative entropy. Analogous
minimizations of arbitrary generalized divergences lead to a new family of
generalized divergences that we call kringel divergences, and for which we
prove various properties including the data-processing inequality.
