A strong Bishop-Phelps property and a new class of Banach spaces with the property $(A)$ of Lindenstrauss

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Abstract: We give a class of bounded closed sets

C

in a Banach space satisfying a generalized and stronger form of the Bishop-Phelps property studied by Bourgain in cite{Bj} for dentable sets. A {it "quantitative version of the Bishop-Phelps-Bollob'as"} will be also given. On the one hand, the density and the risiduality of bounded linear operators attaining their maximum on

C

(known in the literature) will be replaced, for this class of sets, by being the complement of a

s i g m a

-porous set. On the other hand, the result of the paper is applicable for both linear and non-linear operator spaces. When we apply our result to subsets (from this class) whose closed convex hull is the closed unit ball, we obtain a new class of Banach spaces involving property

(A)

introduced by Lindenstrauss. We also establish that this class of Banach spaces is stable under

l^{1}

-sum when the spaces have a same "modulus". Applications to norm attaining mappings in the space of bounded multilinear mappings and the Lipschitz-free space will also be given.

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