Bounded Convex Set Research Articles

QUANTITATIVE CONVERGENCE ANALYSIS OF MULTI-AGENT SYSTEMS

We introduce a characterization of contraction for bounded convex sets. For discrete-time multi-agent systems we provide an explicit upperbound on the rate of convergence to a consensus under the assumptions of contractiveness and (weak) connectedness (across an interval.) Convergence is shown to be exponential when either the system or the function characterizing the contraction is linear.

Closedness of bounded convex sets of asymmetric normed linear spaces and the Hausdorff quasi-metric

If A is a (nonempty) bounded convex subset of an asymmetric normed linear space (X,q), we define the closedness of A as the set clqA clq 1A, and denote by CB0(X) the collection of the closednesses of all (nonempty) bounded convex subsets of (X,q). We show that CB0(X), endowed with the Hausdor quasi-metric of q, can be structured as a quasi-metric cone. Then, {

Harmonic homeomorphisms of the closed disc to itself need be in $W^\{1,p\}$, $p<2$, but not $W^\{1,2\}$

Harmonic maps u from the closed disc onto bounded convex sets of the plane obey u ∈ W 1,p,p < 2.

A ball separation property of bounded convex sets in Hilbert spaces

In this paper we provide a ball separation property of bounded convex sets in a Hilbert space. As a consequence, we obtain a representation form of convex closures and two results about convex functionals.

Continuity properties of the ball hull mapping

The ball hull mapping β associates with each closed bounded convex set K in a Banach space its ball hull β ( K ) , defined as the intersection of all closed balls containing K . We are concerned in this paper with continuity and Lipschitz continuity (with respect to the Hausdorff metric) of the ball hull mapping. It is proved that β is a Lipschitz map in finite dimensional polyhedral spaces. Both properties, finite dimension and polyhedral norm, are necessary for this result. Characterizing the ball hull mapping by means of H -convexity we show, with the help of a remarkable example from combinatorial geometry, that there exist norms with noncontinuous β map, even in finite dimensional spaces. Using this surprising result, we then show that there are infinite dimensional polyhedral spaces (in the usual sense of Klee) for which the map β is not continuous. A property known as ball stability implies that β has Lipschitz constant one. We prove that every Banach space of dimension greater than two can be renormed so that there is an intersection of closed balls for which none of its parallel bodies is an intersection of closed balls, thus lacking ball stability.

A Non-commutative Yosida–Hewitt Theorem and Convex Sets of Measurable Operators Closed Locally in Measure

We present a non-commutative extension of the classical Yosida–Hewitt decomposition of a finitely additive measure into its σ-additive and singular parts. Several applications are given to the characterisation of bounded convex sets in Banach spaces of measurable operators which are closed locally in measure.

On the role of energy convexity in the web function approximation

For a given p > 1 and an open bounded convex set $$\Omega \subset \mathbb{R}^2 ,$$ we consider the minimization problem for the functional $$J_p (u) = \int_\Omega {(\tfrac{1} {p}|\nabla u|^p - u)} $$ over $$W_0^{1,p} (\Omega ).$$ Since the energy of the unique minimizer u p may not be computed explicitly, we restrict the minimization problem to the subspace of web functions, which depend only on the distance from the boundary δΩ. In this case, a representation formula for the unique minimizer v p is available. Hence the problem of estimating the error one makes when approximating J p (u p ) by J p (v p ) arises. When Ω varies among convex bounded sets in the plane, we find an optimal estimate for such error, and we show that it is decreasing and infinitesimal with p. As p → ∞, we also prove that u p − v p converges to zero in $$W_0^{1,m} (\Omega )$$ for all m < ∞. These results reveal that the approximation of minima by means of web functions gains more and more precision as convexity in J p increases.

A characterization of convex calibrable sets in

The main purpose of this paper is to characterize the calibrability of bounded convex sets in by the mean curvature of its boundary, extending the known analogous result in dimension 2. As a by-product of our analysis we prove that any bounded convex set C of class C1,1 has a convex calibrable set K in its interior, and and for any volume V ∈ [|K|,|C|] the solution of the perimeter minimizing problem with fixed volume V in the class of sets contained in C is a convex set. As a consequence we describe the evolution of convex sets in by the minimizing total variation flow.

Analytic Center Cutting-Plane Method with Deep Cuts for Semidefinite Feasibility Problems

An analytic center cutting-plane method with deep cuts for semidefinite feasibility problems is presented. Our objective in these problems is to find a point in a nonempty bounded convex set Γ in the cone of symmetric positive-semidefinite matrices. The cutting plane method achieves this by the following iterative scheme. At each iteration, a query point Ŷ that is an approximate analytic center of the current working set is chosen. We assume that there exists an oracle which either confirms that Ŷ ∈Γ or returns a cut A ∈S m {Y∈S m : A●Y≤ A●YŶ - ξ} ⊃ Γ, where ξ ≥ 0. If Ŷ ∈Γ, an approximate analytic center of the new working set, defined by adding the new cut to the preceding working set, is then computed via a primal Newton procedure. Assuming that Γ contains a ball with radius ∈ > 0, the algorithm obtains eventually a point in Γ, with a worst-case complexity of O *(m 3/∈2) on the total number of cuts generated.

Abelian Theorems, Limit Properties of Conjugate Distributions, and Large Deviations for Sums of Independent Random Vectors

A class of multidimensional absolutely continuous distributions is considered. Each distribution has a moment generating function, which is finite in a bounded convex set S and generates a family of the so-called conjugate distributions. We focus our attention on the limit distributions for this family when the conjugate parameter tends to the boundary of S. As in the one-dimensional case, each limit distribution is obtained as a corollary of the Abel-type theorem. The results obtained are utilized for establishing a local limit theorem for large deviations of arbitrarily high order.

Sharp rate of average decay of the Fourier transform of a bounded set

Estimates for the decay of Fourier transforms of measures have extensive applications in numerous problems in harmonic analysis and convexity including the distribution of lattice points in convex domains, irregularities of distribution, generalized Radon transforms and others. Here we prove that the spherical L 2-average decay rate of the Fourier transform of the Lebesgue measure on an arbitrary bounded convex set in $\mathbb{R}^{d}$ is $${\bigg(\int_{S^{d-1}}{\big|\widehat{\chi}_B(R\omega)\big|}^2d\omega \bigg)}^{{1}/{2}} \lesssim R^{-\frac{d+1}{2}}.\eqno(*)$$ This estimate is optimal for any convex body and in particular it agrees with the familiar estimate for the ball. The above estimate was proved in two dimensions by Podkorytov, and in all dimensions by Varchenko under additional smoothness assumptions. The main result of this paper proves (*) in all dimensions under the convexity hypothesis alone. We also prove that the same result holds if the boundary of ∂Ω is C3/2.

A fixed-point theorem for asymptotically contractive mappings

We present fixed point theorems for a nonexpansive mapping from a closed convex subset of a uniformly convex Banach space into itself under some asymptotic contraction assumptions. They generalize results valid for bounded convex sets or asymptotically compact sets.

Analyse de récession et résultats de stabilité d'une convergence variationnelle, application à la théorie de la dualité en programmation mathématique

Let X be a Banach space and X' its continuous dual. C(X) (resp. C(X')) denotes the set of nonempty convex closed subsets of X (resp. ω*-closed subsets of X') endowed with the topology of uniform convergence of distance functions on bounded sets. This topology reduces to the Hausdorff metric topology on the closed and bounded convex sets [16] and in general has a Hausdorff-like presentation [11]. Moreover, this topology is well suited for estimations and constructive approximations [6-9]. We prove here, that under natural qualification conditions, the stability of the convergence associated to the topology defined on C(X) (resp. C(X')) is preserved by a class of linear transformations. Building on these results, and by identifing each convex function with its epigraph, the stability at the functional level is acquired towards some operations of convex analysis which play a basic role in convex optimization and duality theory. The key hypothesis in the qualification conditions ensuring the functional stability is the notion of inf-local compactness of a convex function introduced in [28] and expressed in the space X' by the quasi-continuity of its conjugate. Then we generalize the stability results of McLinden and Bergstrom [31] and the ones of Beer and Lucchetti [17] in infinite dimension case. Finally we give some applications in convex optimization and mathematical programming in general Banach spaces.

The Analytic Center Cutting Plane Method with Semidefinite Cuts

We analyze an analytic center cutting plane algorithm for convex feasibility problems with semidefinite cuts. The problem of interest seeks a feasible point in a bounded convex set, which contains a full-dimensional ball with $\varepsilon$ $( 0$ is a condition number of the field of cuts.

On weak drop property and quasi-weak drop property

Every weakly sequentially compact convex set in a locally convex space has the weak drop property and every weakly compact convex set has the quasi-weak drop property. An example shows that the quasi-weak drop property is strictly weaker than the weak drop property for closed bounded convex sets in locally convex spaces (even when the spaces are quasi-complete). For closed bounded convex subsets of quasi-complete locally convex spaces, the quasi-weak drop property is equivalent to weak compactness. However, for closed bounded convex sets in sequentially complete locally convex spaces, even the weak drop property does not imply weak compactness. A quasi-complete locally convex space is semi-reexiv e if and only if its closed bounded convex subsets have the quasi-weak drop property. For strong duals of quasi-barrelled spaces, semi-reexivit y is equivalent to every closed bounded convex set having the quasi-weak drop property. From this, reexivit y of a quasi-complete, quasi-barrelled space (in particular, a Fr echet space) is characterized by the quasi-weak drop property of the space and of the strong dual. 1. Introduction. Let (X;kk) be a Banach space and B(X) the closed unit ball fx 2 X : kxk 1g. Given x0 62 B(X), the convex hull of x0 and B(X) is called the drop generated by x0 and denoted by D(x0;B(X)). Dane s (3) proved that in any Banach space (X;kk), for every closed set A at positive distance from B(X), there exists an x02A such that A =fx0g. Modifying the assumption of the Dane s drop theorem, Role- wicz (23) began the study of the drop property for the closed unit ball. He dened the norm kk to have the drop property if for every closed set A disjoint from B(X) there exists an x0 2 A such that D(x0;B(X)) A = fx0g, and he proved that if the norm kk has the drop property then (X;kk) is reexiv e (see (23, Theorem 5)). Giles, Sims and Yorke (7) dened the following weaker variant: the normkk has the weak drop property if for every weakly sequentially closed set A disjoint from B(X), there exists an x02 A such that D(x0;B(X))\A =fx0g, and they showed that this prop- erty is equivalent to (X;kk) being reexiv e. Moreover Kutzarova (11) and

On Summands of Closed Bounded Convex Sets

In this paper properties of the Minkowski-Pontryagin subtraction of closed bounded convex sets are investigated (see Propositions 1-3) and four criteria for summands of closed bounded convex sets are given (see Theorems 1-4).

On the quasi-weak drop property

A new drop property, the quasi-weak drop property, is introduced. Using streaming sequences introduced by Rolewicz, a characterisation of the quasi-weak drop property is given for closed bounded convex sets in a Fréchet space. From this, it is shown that

A QUANTITATIVE VERSION OF A THEOREM DUE TO BORWEIN-REICH-SHAFRIR

We give a quantitative analysis of a result due to Borwein, Reich and Shafrir on the asymptotic behaviour of the general Krasnoselski-Mann iteration for nonexpansive self-mappings of convex sets in arbitrary normed spaces. Besides providing explicit bounds we also get new qualitative results concerning the independence of the rate of asymptotic regularity of that iteration from various input data. In the special case of bounded convex sets, where by well-known results of Ishikawa, Edelstein/O'Brien and Goebel/Kirk the norm of the iteration converges to zero, we obtain uniform bounds which do not depend on the starting point of the iteration and the nonexpansive function, but only depend on the error ε, an upper bound on the diameter of C and some very general information on the sequence of scalars λ k used in the iteration. Only in the special situation, where λ k := λ is constant, uniform bounds were known in that bounded case. For the unbounded case, no quantitative information was known before. Our results were obtained in a case study of analysing non-effective proofs in analysis by certain logical methods. General logical meta-theorems of the author guarantee (at least under some additional restrictions) the extractability of such bounds from proofs of a certain kind and provide an algorithm to extract them. Our results in the present paper (which we present here without any reference to that logical background) were found by applying that method to the original proof of the Borwein/Reich/Shafrir theorem. The general logical method which led to these results will be discussed (with further examples) in [22]. *Basic Research in Computer Science, funded by the Danish National Research Foundation.

On the node fractional iterates

Let M be a bounded convex set of R n , let f : M→ R n be a continuous non-bijective map, we study solutions g of the functional equation g m ( x)= g( g m−1 ( x))= f( x), m⩾2. This functional equation represents the problem of the fractional iteration for f.

A counterexample to the Bishop-Phelps Theorem in complex spaces

The Bishop-Phelps Theorem asserts that the set of functionals which attain the maximum value on a closed bounded convex subsetS of a real Banach spaceX is norm dense inX*. We show that this statement cannot be extended to general complex Banach spaces by constructing a closed bounded convex set with no support points.