### On the extension of rotund norms.

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Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function $f$ defined on a separable Banach space are studied. The conditions are in terms of a majorization of $f$ by a ${C}^{1}$-smooth function, separability of the boundary for $f$ or an approximation of $f$ by Fréchet smooth convex functions.

A class of convex functions where the sets of subdifferentials behave like the unit ball of the dual space of an Asplund space is found. These functions, which we called Asplund functions also possess some stability properties. We also give a sufficient condition for a function to be an Asplund function in terms of the upper-semicontinuity of the subdifferential map.

We investigate the existence of higher order ℓ¹-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space $X=T\left[{(\theta \u2099,\u2099)}_{n=1}^{\infty}\right]$: (1) Every block subspace of X contains an $\ell \xb9{-}_{\omega}$-spreading model, (2) The Bourgain ℓ¹-index ${I}_{b}\left(Y\right)=I\left(Y\right)>{\omega}^{\omega}$ for any block subspace Y of X, (3) $lim\u2098limsup\u2099{\theta}_{m+n}/\theta \u2099>0$ and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these conditions...

Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies $\beta \left(f\right)\le {\omega}^{\xi \u2081}\xb7{\omega}^{\xi \u2082}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1 functions...

A classical theorem of Kuratowski says that every Baire one function on a ${G}_{\delta}$ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a real-valued...

We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $\left({e}_{k}\right)$ is said to be subsequentially minimal if for every normalized block basis $\left({x}_{k}\right)$ of $\left({e}_{k}\right)$, there is a further block basis $\left({y}_{k}\right)$ of $\left({x}_{k}\right)$ such that $\left({y}_{k}\right)$ is equivalent to a subsequence of $\left({e}_{k}\right)$. Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain’s ℓ¹-index are established. It is also shown that a large class of mixed Tsirelson...

We develop a calculus for the oscillation index of Baire one functions using gauges analogous to the modulus of continuity.

* Supported by NSERC (Canada) Let X be a Banach space equipped with norm || · ||. We say that || · || is Gâteaux differentiable at x if for every h ∈ SX(|| · ||), (∗) lim t→0 (||x + th|| − ||x||) / t exists. We say that the norm || · || is Gâteaux differentiable if || · || is Gâteaux differentiable at all x ∈ SX(|| · ||).

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