We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T\left[{({ℳ}_k,{\theta }_k)}_{k=1}^l\right]$ with index $i\left({ℳ}_k\right)$ finite are either $c_0$ or ${\ell }_p$ saturated for some $p$ and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i\left({ℳ}_k\right)$ and the parameter ${\theta }_k$. Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T\left[{({𝒜}_k,{\theta }_k)}_{k=1}^{\infty }\right]$ in terms of the asymptotic behaviour of the sequence $∥{\sum }_{i=1}^n{e}_i∥$ where $\left(e_i\right)$ is the canonical basis....

Let E be an infinite dimensional separable space and for e ∈ E and X a nonempty compact convex subset of E, let qX(e) be the metric antiprojection of e on X. Let n ≥ 2 be an arbitrary integer. It is shown that for a typical (in the sence of the Baire category) compact convex set X ⊂ E the metric antiprojection qX(e) has cardinality at least n for every e in a dense subset of E.

We give a new construction of uniformly convex norms with a power type modulus on super-reflexive spaces based on the notion of dentability index. Furthermore, we prove that if the Szlenk index of a Banach space is less than or equal to ω (first infinite ordinal) then there is an equivalent weak* lower semicontinuous positively homogeneous functional on X* satisfying the uniform Kadec-Klee Property for the weak*-topology (UKK*). Then we solve the UKK or UKK* renorming problems for Lp(X) spaces...

Every separable Banach space with $C^{\left(n\right)}$-smooth norm (Lipschitz bump function) admits an equivalent norm (a Lipschitz bump function) which is both uniformly Gâteaux smooth and $C^{\left(n\right)}$-smooth. If a Banach space admits a uniformly Gâteaux smooth bump function, then it admits an equivalent uniformly Gâteaux smooth norm.

Let 1 ≤ p < 2 and let $L_p=L_p[0,1]$ be the classical $L_p$-space of all (classes of) p-integrable functions on [0,1]. It is known that a sequence of independent copies of a mean zero random variable $f\in L_p$ spans in $L_p$ a subspace isomorphic to some Orlicz sequence space $l_M$. We give precise connections between M and f and establish conditions under which the distribution of a random variable $f\in L_p$ whose independent copies span $l_M$ in $L_p$ is essentially unique.

The purpose of this paper is to continue the investigations on the homothety of unit balls and isoperimetrices in higher-dimensional Minkowski spaces for the Holmes-Thompson measure and the Busemann measure. Moreover, we show a strong relation between affine isoperimetric inequalities and Minkowski geometry by proving some new related inequalities.

Given a Young function $\Phi$, we study the existence of copies of $c_0$ and ${\ell }_{\infty }$ in ${\mathrm{c}abv}_{\Phi }(\mu ,X)$ and in ${\mathrm{c}absv}_{\Phi }(\mu ,X)$, the countably additive, $\mu$-continuous, and $X$-valued measure spaces of bounded $\Phi$-variation and bounded $\Phi$-semivariation, respectively.

Every weakly sequentially compact convex set in a locally convex space has the weak drop property and every weakly compact convex set has the quasi-weak drop property. An example shows that the quasi-weak drop property is strictly weaker than the weak drop property for closed bounded convex sets in locally convex spaces (even when the spaces are quasi-complete). For closed bounded convex subsets of quasi-complete locally convex spaces, the quasi-weak drop property is equivalent to weak compactness....

We show that every infinite-dimensional Banach space with separable dual admits an equivalent norm which is weakly locally uniformly rotund but not locally uniformly rotund.

In this note we study the topological structure of weighted James spaces J(h). In particular we prove that J(h) is isomorphic to J if and only if the weight h is bounded. We also provide a description of J(h) if the weight is a non-decreasing sequence.