### Reflexive Banach spaces without equivalent norms which are uniformly convex or uniformly differentiable in every direction

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A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences ${\left({x}_{j}^{i}\right)}_{j=1}^{\infty}\subseteq X$, 1 ≤ i ≤ m, for all permutations σ of 1,...,m and all ultrafilters ₁,...,ₘ on ℕ, $li{m}_{n\u2081,\u2081}...li{m}_{n\u2098,\u2098}\left|\right|{\sum}_{i=1}^{m}{x}_{{n}_{i}}^{i}\left|\right|\le Cli{m}_{{n}_{\sigma \left(1\right)}{,}_{\sigma \left(1\right)}}...li{m}_{{n}_{\sigma \left(m\right)}{,}_{\sigma \left(m\right)}}\left|\right|{\sum}_{i=1}^{m}{x}_{{n}_{i}}^{i}\left|\right|$. We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences ${\left({x}_{j}^{i}\right)}_{j=1}^{\infty}$. Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then all spreading...

In this article, we consider the (weak) drop property, weak property (a), and property (w) for closed convex sets. Here we give some relations between those properties. Particularly, we prove that C has (weak) property (a) if and only if the subdifferential mapping of Cº is (n-n) (respectively, (n-w)) upper semicontinuous and (weak) compact valued. This gives an extension of a theorem of Giles and the first author.

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