### $\u2102$-convexity in infinite-dimensional Banach spaces and applications to Kergin interpolation.

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Hagler and the first named author introduced a class of hereditarily ${l}_{1}$ Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily ${l}_{p}$ Banach spaces for $1\le p<\infty $. Here we use these spaces to introduce a new class of hereditarily ${l}_{p}\left({c}_{0}\right)$ Banach spaces analogous of the space of Popov. In particular, for $p=1$ the spaces are further examples of hereditarily ${l}_{1}$ Banach spaces failing the Schur property.

When the set of closed subspaces of C(Δ), where Δ is the Cantor set, is equipped with the standard Effros-Borel structure, the graph of the basic relations between Banach spaces (isomorphism, being isomorphic to a subspace, quotient, direct sum,...) is analytic non-Borel. Many natural families of Banach spaces (such as reflexive spaces, spaces not containing ℓ₁(ω),...) are coanalytic non-Borel. Some natural ranks (rank of embedding, Szlenk indices) are shown to be coanalytic ranks. Applications...

It is shown that if 𝓢 is a commuting family of weak* continuous nonexpansive mappings acting on a weak* compact convex subset C of the dual Banach space E, then the set of common fixed points of 𝓢 is a nonempty nonexpansive retract of C. This partially solves an open problem in metric fixed point theory in the case of commutative semigroups.

We compare the yields of two methods to obtain Bernstein type pointwise estimates for the derivative of a multivariate polynomial in a domain where the polynomial is assumed to have sup norm at most 1. One method, due to Sarantopoulos, relies on inscribing ellipses in a convex domain K. The other, pluripotential-theoretic approach, mainly due to Baran, works for even more general sets, and uses the pluricomplex Green function (the Zaharjuta-Siciak extremal function). When the inscribed ellipse method...

A family is constructed of cardinality equal to the continuum, whose members are totally incomparable hereditarily indecomposable Banach spaces.

We prove the existence of a contractive mapping on a weakly compact convex set in a Banach space that is fixed point free. This answers a long-standing open question.

We show that the Schreier sets ${S}_{\alpha}(\alpha <{\omega}_{1})$ have the following dichotomy property. For every hereditary collection ℱ of finite subsets of ℱ, either there exists infinite $M={\left({m}_{i}\right)}_{i=1}^{\infty}\subseteq \mathbb{N}$ such that ${S}_{\alpha}\left(M\right)={m}_{i}:i\in E:E\in {S}_{\alpha}\subseteq \mathcal{F}$, or there exist infinite $M={\left({m}_{i}\right)}_{i=1}^{\infty},N\subseteq \mathbb{N}$ such that $\mathcal{F}\left[N\right]\left(M\right)={m}_{i}:i\in F:F\in \mathcal{F}andF\subset N\subseteq {S}_{\alpha}$.

In this paper, a precise projection decomposition in reflexive, smooth and strictly convex Orlicz-Bochner spaces is given by the representation of the duality mapping. As an application, a representation of the metric projection operator on a closed hyperplane is presented.