Diameter two properties, convexity and smoothness
TA Abrahamsen, V Lima, O Nygaard… - Milan Journal of …, 2016 - Springer
Milan Journal of Mathematics, 2016•Springer
We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence
of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual
from being strictly convex and being smooth, and we initiate the investigation whether the
same is true for the (local) diameter 2 property. We also give characterizations of the
following property for a Banach space XX:“For every slice SS of B_X BX and every norm-one
element xx in SS, there is a point y ∈ S y∈ S in distance as close to 2 as we want.” Spaces …
of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual
from being strictly convex and being smooth, and we initiate the investigation whether the
same is true for the (local) diameter 2 property. We also give characterizations of the
following property for a Banach space XX:“For every slice SS of B_X BX and every norm-one
element xx in SS, there is a point y ∈ S y∈ S in distance as close to 2 as we want.” Spaces …
Abstract
We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties.We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space : “For every slice of and every norm-one element in , there is a point in distance as close to 2 as we want.” Spaces with this property are shown to have non-smooth bidual.
Springer
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