  We present a systematic study of the geometric R\'enyi divergence (GRD), also
known as the maximal R\'enyi divergence, from the point of view of quantum
information theory. We show that this divergence, together with its extension
to channels, has many appealing structural properties. For example we prove a
chain rule inequality that immediately implies the "amortization collapse" for
the geometric R\'enyi divergence, addressing an open question by Berta et al.
[arXiv:1808.01498, Equation (55)] in the area of quantum channel
discrimination. As applications, we explore various channel capacity problems
and construct new channel information measures based on the geometric R\'enyi
divergence, sharpening the previously best-known bounds based on the
max-relative entropy while still keeping the new bounds single-letter
efficiently computable. A plethora of examples are investigated and the
improvements are evident for almost all cases.
