In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continous convex functions on a vector space R and vector-valued functions in a weakly compact subset of a Banach vector space generated by m Lμ-spaces for 1 ≤ p < +∞. Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach...

A k-set of a finite set S of points in the plane is a subset of cardinality k that can be separated from the rest by a straight line. The question of how many k-sets a set of n points can contain is a long-standing open problem where a lower bound of (n logk) and an upper bound of O(nk1/3) are known today. Under certain restrictions on the set S, for example, if all points lie on a convex curv...

We study the asymptotic behavior of the sequence of the iterates for a nonexpansive mapping, defined on a suitable subset of a Banach space with Opial's condition. Some results are stated also for semigroups of nonexpansive mappings and for mappings of asymptotically nonexpansive type in uniformly convex Banach spaces with Opial's condition.

Let X be a complex Banach space, a norming set for X , and D ⊂ X a bounded, closed, and convex domain such that its norm closure D is compact in σ(X , ). Let ∅ = C ⊂D lie strictly inside D. We study convergence properties of infinite products of those selfmappings of C which can be extended to holomorphic self-mappings of D. Endowing the space of sequences of such mappings with an appropriate m...

The topology of the weighted inductive limit of Fréchet spaces of entire functions in N variables which is obtained as the Fourier Laplace transform of the space of quasianalytic functionals on an open convex subset of R cannot be described by means of weighted supseminorms.

Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let {Ti}i=1 be N nonexpansive self-mappings of K with F = ∩i=1F (Ti) ̸= ∅ (here F (Ti) denotes the set of fixed points of Ti). Suppose that one of the mappings in {Ti}i=1 is semi-compact. Let {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1) and {βn} ⊂ [τ, 1] for some τ ∈ (0, 1]. For arbitrary x0 ∈ K, let the sequence {x...

In a complete simply connected Riemannian manifold X of pinched negative curvature, we give a sharp criterion for a subset C to be the ǫ-neighbourhood of some convex subset of X, in terms of the extrinsic curvatures of the boundary of C. 1

The aim of this paper is to present some existence results for nearest-point and farthestpoint problems, in connection with some geometric properties of Banach spaces. The idea goes back to Efimov and Stečkin who, in a series of papers (see [28, 29, 30, 31]), realized for the first time that some geometric properties of Banach spaces, such as strict convexity, uniform convexity, reflexivity, an...

Let E be a subset of a convex, open, bounded, planar set G. Let P (E, G) be the relative perimeter of E (the length of the boundary of E contained in G). We obtain relative geometric inequalities comparing the relative perimeter of E with the relative diameter of E and with its relative inradius. We prove the existence of both extremal sets and maximizers for these inequalities and describe the...

In this paper we present inequalities for integrals of functions that are the composition of nonnegative convex functions on an open convex set of a vector space R and vectorvalued functions in a weakly compact subset of a Banach vector space generated by m Lμ-spaces for 1 ≤ p < +∞ and inequalities when these vector-valued functions are in a weakly* compact subset of a Banach vector space gener...