We prove a stability result on the minimal self-perimeter L(B) of the unit disk B of a normed plane: if L(B) = 6 + ε for a sufficiently small ε, then there exists an affinely regular hexagon S such that S ⊂ B ⊂ (1 + 6∛ε) S.

The problem of strict convexity of the Besicovitch-Orlicz space of almost periodic functions is considered here in connection with the Orlicz norm. We give necessary and sufficient conditions in terms of the function f generating the space.

We prove that there exists an absolute constant c such that for any positive integer n and any system Φ of $2^n$ characters of a compact abelian group, $2^{-n/2}{t}_{\Phi }\left(T\right)\le c{n}^{-1/2}{t}_n\left(T\right)$, where T is an arbitrary operator between Banach spaces, $t_{\Phi }\left(T\right)$ is the type norm of T with respect to Φ and $t_n\left(T\right)$ is the usual Rademacher type-2 norm computed with n vectors. For the system of the first $2^n$ Walsh functions this is even true with c=1. This result combined with known properties of such type norms provides easy access to quantitative versions of...

In [5], we characterized the uniform convexity with respect to the Luxemburg norm of the Besicovitch-Orlicz space of almost periodic functions. Here we give an analogous result when this space is endowed with the Orlicz norm.

We prove that in Orlicz spaces endowed with Orlicz norm the uniformly normal structure is equivalent to the reflexivity.

In this note we show that if the ratio of the minimal volume V of n-dimensional parallelepipeds containing the unit ball of an n-dimensional real normed space X to the maximal volume v of n-dimensional crosspolytopes inscribed in this ball is equal to n!, then the relation of orthogonality in X is symmetric. Hence we deduce the following properties: (i) if V/v=n! and if n>2, then X is an inner product space; (ii) in every finite-dimensional normed space there exist at least two different Auerbach...

We obtain the necessary and sufficient condition of weak star uniformly rotund point in Orlicz spaces.

The concept of WM point is introduced and the criterion of WM property in Orlicz function spaces endowed with Luxemburg norm is given.

In this paper, we introduce the concept of WM point and obtain the criterion of WM points for Orlicz function spaces endowed with Orlicz norm and the criterion of WM property for Orlicz space.

We obtain the criterion of the WM property for Orlicz sequence spaces endowed with the Orlicz norm.

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...