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Potentially elementary question on affine functions on Banach spaces

In Measures Which Agree On Balls by Hoffmann-Jørgensen, it is claimed that the function defined on $T(x)$, the set of normals to the unit sphere at $x$, given by $ \varphi(x^*) = \left\{ \begin{array} …
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1 vote
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Gateaux differentiability of the norm in Banach spaces

I'm struggling to understand a particular implication in the proof of Corollary 5 of this paper involving Gateaux differentiability of the norm. The claim is that Gateaux differentiability of the norm …
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