  We show that if $x$ is a strongly extreme point of a bounded closed convex
subset of a Banach space and the identity has a geometrically and topologically
good enough local approximation at $x$, then $x$ is already a denting point. It
turns out that such an approximation of the identity exists at any strongly
extreme point of the unit ball of a Banach space with the unconditional compact
approximation property. We also prove that every Banach space with a Schauder
basis can be equivalently renormed to satisfy the sufficient conditions
mentioned. In contrast to the above results we also construct a non-symmetric
norm on $c_0$ for which all points on the unit sphere are strongly extreme, but
none of these points are denting.
