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On the size of approximately convex sets in normed spaces

S. Dilworth, Ralph Howard, James Roberts (2000)

Studia Mathematica

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 ℋ ( A , C o ( A ) ) ≥ l o g 2 n - 1 and d i a m ( A ) ≤ C √ n ( l n n ) 2 , 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 statistical and σ-cores

Hüsamettın Çoşkun, Celal Çakan, Mursaleen (2003)

Studia Mathematica

In [11] and [7], the concepts of σ-core and statistical core of a bounded number sequence x have been introduced and also some inequalities which are analogues of Knopp’s core theorem have been proved. In this paper, we characterize the matrices of the class ( S ∩ m , V σ ) r e g and determine necessary and sufficient conditions for a matrix A to satisfy σ-core(Ax) ⊆ st-core(x) for all x ∈ m.

On the strong McShane integral of functions with values in a Banach space

Štefan Schwabik, Ye Guoju (2001)

Czechoslovak Mathematical Journal

The classical Bochner integral is compared with the McShane concept of integration based on Riemann type integral sums. It turns out that the Bochner integrable functions form a proper subclass of the set of functions which are McShane integrable provided the Banach space to which the values of functions belong is infinite-dimensional. The Bochner integrable functions are characterized by using gauge techniques. The situation is different in the case of finite-dimensional valued vector functions....

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