Bounded Convex Subset Research Articles

Extremal structure in ultrapowers of Banach spaces

Given a bounded convex subset C of a Banach space X and a free ultrafilter {mathcal {U}}, we study which points (x_i)_{mathcal {U}} are extreme points of the ultrapower C_{mathcal {U}} in X_{mathcal {U}}. In general, we obtain that when {x_i} is made of extreme points (respectively denting points, strongly exposed points) and they satisfy some kind of uniformity, then (x_i)_{mathcal {U}} is an extreme point (respectively denting point, strongly exposed point) of C_mathcal U. We also show that every extreme point of C_{{mathcal {U}}} is strongly extreme, and that every point exposed by a functional in (X^*)_{{mathcal {U}}} is strongly exposed, provided that mathcal U is a countably incomplete ultrafilter. Finally, we analyse the extremal structure of C_{mathcal {U}} in the case that C is a super weakly compact or uniformly convex set.

Which set is the numerical range of an operator?

In this survey paper, we consider the problem of which nonempty bounded convex subset of the complex plane is the numerical range of some bounded linear operator on a complex separable Hilbert space. We start in Section 1 with general operators and move subsequently to operators in certain special classes such as normal and hyponormal operators in Section 2, Toeplitz operators in Section 3, Hankel operators in Section 4, compact operators in Section 5, nilpotent operators and roots of identity in Section 6, Sn-matrices and companion matrices in Section 7, and, finally, nonnegative matrices in Section 8. The known main results will be briefly sketched, which are interspersed with relevant unsolved problems.

Consensus in the Hegselmann–Krause Model

This paper is concerned with the probability of consensus in a multivariate, spatially explicit version of the Hegselmann-Krause model for the dynamics of opinions. Individuals are located on the vertices of a finite connected graph representing a social network, and are characterized by their opinion, with the set of opinions $\Delta$ being a general bounded convex subset of a finite dimensional normed vector space. Having a confidence threshold $\tau$, two individuals are said to be compatible if the distance (induced by the norm) between their opinions does not exceed the threshold $\tau$. Each vertex $x$ updates its opinion at rate the number of its compatible neighbors on the social network, which results in the opinion at $x$ to be replaced by a convex combination of the opinion at $x$ and the nearby opinions: $\alpha$ times the opinion at $x$ plus $(1 - \alpha)$ times the average opinion of its compatible neighbors. The main objective is to derive a lower bound for the probability of consensus when the opinions are initially independent and identically distributed with values in the opinion set $\Delta$.

Some fixed point results for (c)-mappings in Banach spaces

In this paper, we solve two fixed point problems associated with the class of (c)-mappings. The first one is devoted to obtain the existence of fixed points for such mappings defined on $$\hbox {weak}^{\star }$$ closed bounded convex subsets of duals of separable Banach spaces when the orthogonality relation $$\perp $$ is uniformly $$\hbox {weak}^{\star }$$ approximately symmetric. For the second problem, using the idea of R. Smarzewski (On firmly nonexpansive mappings, Proc. Am. Math. Soc 113(3):723–725,1991), we prove the existence of fixed points for such mappings which are defined on a finite union of weakly compact convex subsets of UCED (uniformly convex in every direction) Banach spaces.

A fixed-point characterization of weakly compact sets in L1(μ) spaces

Let (Ω,Σ,μ) be a σ-finite measure space and consider the Lebesgue function space L1(μ) endowed with its standard norm. We obtain a characterization of weak compactness for closed bounded convex subsets of L1(μ) in terms of the existence of fixed points for certain classes of eventually affine, uniformly Lipschitzian mappings.

Probability of consensus in the multivariate Deffuant model on finite connected graphs

The Deffuant model is a spatial stochastic model for the dynamics of opinions in which individuals are located on a connected graph representing a social network and characterized by a number in the unit interval representing their opinion. The system evolves according to the following averaging procedure: at each time step, two neighbors are randomly chosen and interact if and only if the distance between their opinions does not exceed a certain confidence threshold, with each interaction resulting in the neighbors’ opinions getting closer to each other. Most of the analytical results established so far about this model assume that the individuals are located on the integers. In contrast, we study the more realistic case where the social network can be any finite connected graph. In addition, we extend the opinion space to any bounded convex subset of a normed vector space where the norm is used to measure the level of disagreement or distance between the opinions. Our main result gives a lower bound for the probability of consensus. Our proof leads to a universal lower bound that depends on the confidence threshold, the opinion space (convex subset and norm) and the initial distribution, but not on the size or the topology of the social network.

On topological groups with an approximate fixed point property.

A topological group G has the Approximate Fixed Point (AFP) property on a bounded convex subset C of a locally convex space if every continuous affine action of G on C admits a net ( x i ) , x i ∈ C , such that x i - g ⁢ x i ⟶ 0 for all g ∈ G . In this work, we study the relationship between this property and amenability.

Inner γ-Convex Functions in Normed Linear Spaces

A real-valued function f defined on a convex subset D of some normed linear space X is said to be inner γ-convex w.r.t. some fixed roughness degree γ > 0 if there is a ν ∈ [0, 1] such that $$\sup \limits _{\lambda \in [2, 1+1/\nu ]} \left (f((1-\lambda )x_{0}+\lambda x_{1})- (1-\lambda )f(x_{0})- \lambda f(x_{1})\right )\geq 0 $$ holds for all x 0, x 1 ∈ D satisfying ∥x 0 − x 1∥ = ν γ and −(1/ν)x 0 + (1 + 1/ν)x 1 ∈ D. The requirement of this kind of roughly generalized convex functions is very weak; nevertheless, they also possess properties similar to those of convex functions relative to their supremum. For instance, if an inner γ-convex function defined on some bounded convex subset D of an inner product space attains its maximum, then it has maximizers at some strictly γ-extreme points of D. In this paper, some sufficient conditions and examples for γ-convex functions and several properties relative to the location of their maximizers are given.

Equivariant absolute extensor property on hyperspaces of convex sets

Let G be a compact group acting on a Banach space L by means of affine transformations. The action of G on L induces a natural continuous action on cc(L), the hyperspace of all compact convex subsets of L endowed with the Hausdorff metric topology. The main result of this paper states that the G-space cc(L) is a G-AE. Under some extra assumptions, this result can be extended to CB(L), the hyperspace of all closed and bounded convex subsets of L.

On some relationships between biorthogonal systems, linear topologies and the weak‐AFPP

In this paper, we obtain new results for the weak‐AFPP in abstract spaces by exploiting biorthogonal systems techniques. Firstly, we investigate the strong‐AFPP on countably infinite dimensional Hausdorff locally convex spaces. Spaces of this class are shown to be sequentially complete iff they have the hereditary FPP for totally bounded, closed convex sets. This might open a research line for the analysis of weak‐AFPP in such frames. In connection, we provide a simple criterion for the containement of ℓ1‐sequences in terms of strongly‐equicontinuous biorthogonal systems. We then establish a few results concerning the existence of Hausdorff finer vector topologies on abstract spaces having as prescribed condition the existence of such systems. The proofs are based on methods of Peck and Porta concerning building of finer vector topologies, and a classical construction of Singer which allows us to prove under rather natural conditions the existence of equicontinuous biorthogonal systems in metrizable locally convex spaces. These results are compatible with the failure of the weak‐AFPP. We also study the inverse problem by proving that every infinite dimensional vector space admits a (non‐locally convex) Hausdorff vector topology which is complete, non‐metrizable and is compatible with a bounded Hamel Schauder basis. It is shown further that such a topology has the ‐AFPP, where is the linear span of coefficient functionals associated to a Hamel basis. Finally, inspired by a result of Shapiro, we observe that if X is a non‐locally convex F‐space with an absolute basis, then the weak‐AFPP is equivalent to the fact that every bounded convex subset of X is compact.

Optimal approximate fixed point results in locally convex spaces

Let C be a convex subset of a locally convex space. We provide optimal approximate fixed point results for sequentially continuous maps f:C→C¯. First, we prove that, if f(C) is totally bounded, then it has an approximate fixed point net. Next, it is shown that, if C is bounded but not totally bounded, then there is a uniformly continuous map f:C→C without approximate fixed point nets. We also exhibit an example of a sequentially continuous map defined on a compact convex set with no approximate fixed point sequence. In contrast, it is observed that every affine (not-necessarily continuous) self-mapping of a bounded convex subset of a topological vector space has an approximate fixed point sequence. Moreover, we construct an affine sequentially continuous map from a compact convex set into itself without fixed points.

Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of linear parabolic initial-and final boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary dimension. Under certain conditions regarding their endpoint distributions, we also prove that such processes become reversible Markov diffusions. Furthermore we show that those diffusions satisfy two It\^o equations for some suitably constructed Wiener processes, and from that analysis derive Feynman-Kac representations for the solutions to the given equations. We then illustrate some of our results by considering the heat equation with Neumann boundary conditions both in a one-dimensional bounded interval and in a two-dimensional disk.

Steepest descent curves of convex functions on surfaces of constant curvature

Let S be a complete surface of constant curvature K = ±1, i.e., S2 or л2, and Ω ⊂ S a bounded convex subset. If S = S2, assume also diameter(Ω) < π/2. It is proved that the length of any steepest descent curve of a quasi-convex function in Ω is less than or equal to the perimeter of Ω. This upper bound is actually proved for the class of G-curves, a family of curves that naturally includes all steepest descent curves. In case S = S2, the existence of G-curves, whose length is equal to the perimeter of their convex hull, is also proved, showing that the above estimate is indeed optimal. The results generalize theorems by Manselli and Pucci on steepest descent curves in the Euclidean plane.

Fixed point properties of semigroups of nonexpansive mappings

We prove that if H is a Hilbert space then the Schatten (trace) class operators on H has the weak ∗ fixed point property for left reversible semigroups. This answered positively a problem raised by A.T.-M. Lau. We also prove that if M is a finite von Neumann algebra then any nonempty bounded convex subset of the non-commutative L 1 -space associated to M that is compact for the measure topology has the fixed point property for left reversible semigroups.

The Minkowski problem for the torsional rigidity

We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of ℝ n . For the existence part we apply the variational method introduced by Jerison in: Adv. Math. 122 (1996), pp. 262-279. Uniqueness follows from the Brunn-Minkowski inequality for the torsional rigidity and corresponding equality conditions.

Best proximity pair theorems for relatively nonexpansive mappings

Let A, B be nonempty closed bounded convex subsets of a uniformly convex Banach space and T : A∪B → A∪B be a map such that T(A) ⊆ B, T(B) ⊆ A and ǁTx − Tyǁ ≤ ǁx − yǁ, for x in A and y in B. The fixed point equation Tx = x does not possess a solution when A ∩ B = Ø. In such a situation it is natural to explore to find an element x0 in A satisfying ǁx0 − Tx0ǁ = inf{ǁa − bǁ : a ∈ A, b ∈ B} = dist(A,B). Using Zorn’s lemma, Eldred et.al proved that such a point x0 exists in a uniformly convex Banach space settings under the conditions stated above. In this paper, by using a convergence theorem we attempt to prove the existence of such a point x0 (called best proximity point) without invoking Zorn’s lemma.

Coverings of Banach spaces: beyond the Corson theorem

A well-known result due to H. Corson has been recently improved by the authors. In its final form it essentially reads as follows: for any covering τ by closed bounded convex subsets of any Banach space X containing a separable infinite-dimensional dual space, a (algebraically) finite-dimensional compact set C can always be found that meets infinitely many members of τ. In the present paper we investigate how small the dimension of this compact set can be, in the case the members of τ are closed bounded convex bodies satisfying general conditions of rotundity or smoothness type. In particular, such a compact set turns out to be a segment whenever the members of τ are rotund or smooth bodies in the usual sense.

A note on the uniform approximation of continuous affine functions

It is known that if X is a compact convex subset of a locally convex space E, then the set of all continuous affine functions on X equals the set ( E * + R ) | X ¯ . We study the possibility of extending this theorem for noncompact sets. For a bounded convex subset X of a locally convex space E we characterize those continuous affine functions belonging to ( E * + R ) | X ¯ . We also give an example of a closed bounded convex subset X of c 0 and a continuous affine function on X, which does not belong to ( l 1 + R ) | X ¯ .

Finitely locally finite coverings of Banach spaces

A well-known result due to H. Corson states that, for any covering τ by closed bounded convex subsets of any Banach space X containing an infinite-dimensional reflexive subspace, there exists a compact subset C of X that meets infinitely many members of τ. We strengthen this result proving that, even under the weaker assumption that X contains an infinite-dimensional separable dual space, an (algebraically) finite-dimensional compact set C with that property can always be found.

Norm-preserving extensions of functionals and denting points of convex sets

Let W and Z be Banach spaces, and let \(X\subset W\) and \(Y\subset Z\) be closed subspaces. Let \(\mathcal{L}\) be a subspace of \(\mathcal{L}(W^*, Z^{**})\), the Banach space of bounded linear operators from W* to Z**, containing \(X\otimes Y\). We describe, for \(x^{*}\in X^{*}\) and \(y^*\in Y^*\), all norm-preserving extensions of \(x^{*}\otimes y^{*}\in (X\otimes Y)^\ast\) to the space \(\mathcal{L}\) in terms of convergence of convex combinations. We also characterize denting points of bounded convex subsets of Banach spaces in similar terms. Various applications are presented.