  We develop tools to produce equivalent norms with specific local geometry
around infinitely many points in the sphere of a Banach space via an inductive
procedure. We combine this process with smoothness results and techniques to
solve two open problems posed in the recently published monograph [GMZ22] by A.
J. Guirao, V. Montesinos and V. Zizler. Specifically, on the one hand we
construct in every separable Banach space admitting a $C^k$-smooth norm an
equivalent norm which is $C^k$-smooth but fails to be uniformly G\^ateaux in
any direction; and on the other hand we produce in $c_0(\Gamma)$ for any
infinite $\Gamma$ a $C^\infty$-smooth norm whose ball is dentable but whose
sphere lacks any extreme points.
