Bishop-Phelps-Bolloba's theorem on bounded closed convex sets

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This paper deals with the Bishop-Phelps-Bollobás property (BPBp for short) on bounded closed convex subsets of a Banach space X, not just on its closed unit ball Bₓ. We firstly prove that the BPBp holds for bounded linear functionals on arbitrary bounded closed convex subsets of a real Banach space. We show that for all finite dimensional Banach spaces X and Y the pair (X,Y) has the BPBp on every bounded closed convex subset D of X, and also that for a Banach space Y with property (β) the pair (X,Y) has the BPBp on every bounded closed absolutely convex subset D of an arbitrary Banach space X. For a bounded closed absorbing convex subset D of X with positive modulus convexity we get that the pair (X,Y) has the BPBp on D for every Banach space Y. We further obtain that for an Asplund space X and for a locally compact Hausdorff L, the pair (X, C₀(L)) has the BPBp on every bounded closed absolutely convex subset D of X. Finally we study the stability of the BPBp on a bounded closed convex set for the ℓ₁-sum or ℓ∞-sum of a family of Banach spaces.