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

Renormings preserving local geometry at countably many points in spheres of Banach spaces and applications