Daugavet- and Delta-points in spaces of Lipschitz functions

A norm one element $x$ of a Banach space is a Daugavet-point (respectively,~a $Δ$-point) if every slice of the unit ball (respectively,~every slice of the unit ball containing $x$) contains an element that is almost at distance 2 from $x$. We prove the equivalence of Daugavet- and $Δ$-points in spaces of Lipschitz functions over proper metric spaces and provide two characterizations for them. Furthermore, we show that in some spaces of Lipschitz functions, there exist $Δ$-points that are not Daugavet-points. Lastly, we prove that every space of Lipschitz functions over an infinite metric space contains a $Δ$-point but might not contain any Daugavet-points.

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