On the relation of the bounded approximation property and a finite dimensional decomposition in nuclear Fréchet spaces
On the Relationship Between yp-Radonifying Operators and Other Operator Ideals in Banach Spaces of Stable Type p.
On the Riesz-Fischer theorem in a smooth Banach space
On the size of approximately convex sets in normed spaces
Let X be a normed space. A set A ⊆ X is approximately convexif d(ta+(1-t)b,A)≤1 for all a,b ∈ A and t ∈ [0,1]. We prove that every n-dimensional normed space contains approximately convex sets A with and , where ℋ denotes the Hausdorff distance. These estimates are reasonably sharp. For every D>0, we construct worst possible approximately convex sets in C[0,1] such that ℋ(A,Co(A))=(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.
On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space
We study the size of the sets of gradients of bump functions on the Hilbert space , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space can be uniformly approximated by smooth Lipschitz functions so that the cones generated by the ranges of its derivatives have empty interior. This implies that there are smooth Lipschitz bumps...
On the space of -summable sequences
On the stability of the fixed point property in spaces.
On the stability of the unit circle with minimal self-perimeter in normed planes
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.
On the strict convexity of the Besicovitch-Orlicz space of almost periodic functions with Orlicz norm.
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.
On the structure of Banach spaces with Mazur's intersection property.
On the type constants with respect to systems of characters of a compact abelian group
We prove that there exists an absolute constant c such that for any positive integer n and any system Φ of characters of a compact abelian group, , where T is an arbitrary operator between Banach spaces, is the type norm of T with respect to Φ and is the usual Rademacher type-2 norm computed with n vectors. For the system of the first 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...
On the uniform convexity of the Besicovitch-Orlicz space of almost periodic functions with Orlicz norm
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.
On the uniformly normal structure of Orlicz spaces with Orlicz norm
We prove that in Orlicz spaces endowed with Orlicz norm the uniformly normal structure is equivalent to the reflexivity.
On the volume method in the study of Auerbach bases of finite-dimensional normed spaces
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...
On the volume of unit balls in Banach spaces
On the weak star uniformly rotund points of Orlicz spaces.
We obtain the necessary and sufficient condition of weak star uniformly rotund point in Orlicz spaces.
On the weak uniform rotundity of Banach spaces.
On the WM points of Orlicz function spaces endowed with Luxemburg norm.
The concept of WM point is introduced and the criterion of WM property in Orlicz function spaces endowed with Luxemburg norm is given.
On the WM points of Orlicz function spaces endowed with Orlicz norm.
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.
On the WM property of Orlicz sequence spaces endowed with the Orlicz norm
We obtain the criterion of the WM property for Orlicz sequence spaces endowed with the Orlicz norm.