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Antoni Machowski

Jagiellonian University 

Strongly exposing functionals of convex weakly compact sets

[27.02.2024, 12.15. THE MEETING IS HELD HYBRID, STATIONARY PART in 1016]

Abstract: For a closed, convex bounded subset C of a Banach space X we call a point x in C strongly exposed if some functional on X attains its strong maximum in x. Functionals of norm one that satisfy the above condition for some point in C are called strongly exposing functionals of C. In my talk I will present a proof by J. Bourgain of celebrated results by J. Lindenstrauss and S.L. Troyanski stating that a convex and weakly compact subset of a Banach space is the closed convex hull of its strongly exposed points and the set of strongly exposing functionals of C is a dense G-delta subset of the unit sphere of X*.


J. Lindenstrauss, On operators which attain their norm (1963)

S. L. Trojanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces (1970)

J. Bourgain, Strongly exposed points in weakly compact convex sets in Banach spaces (1976)

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