Class Of Convex Sets Research Articles

An isoperimetric inequality for the first Steklov–Dirichlet Laplacian eigenvalue of convex sets with a spherical hole

In this paper we prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint. More precisely, if $\Omega=\Omega_0 \setminus \bar{B}_{R_1}$, where $B_{R_1}$ is the ball centered at the origin with radius $R_1>0$ and $\Omega_0\subset\mathbb{R}^n$, $n\geq 2$, is an open bounded and convex set such that $B_{R_1}\Subset \Omega_0$, then the first Steklov-Dirichlet eigenvalue $\sigma_1(\Omega)$ has a maximum when $R_1$ and the measure of $\Omega$ are fixed. Moreover, if $\Omega_0$ is contained in a suitable ball, we prove that the spherical shell is the maximum.

Cube root weak convergence of empirical estimators of a density level set

Given n independent random vectors with common density f on Rd, we study the weak convergence of three empirical-measure based estimators of the convex λ-level set Lλ of f, namely the excess mass set, the minimum volume set and the maximum probability set, all selected from a class of convex sets A that contains Lλ. Since these set-valued estimators approach Lλ, even the formulation of their weak convergence is nonstandard. We identify the joint limiting distribution of the symmetric difference of Lλ and each of the three estimators, at rate n−1/3. It turns out that the minimum volume set and the maximum probability set estimators are asymptotically indistinguishable, whereas the excess mass set estimator exhibits “richer” limit behavior. Arguments rely on the boundary local empirical process, its cylinder representation, dimension-free concentration around the boundary of Lλ, and the set-valued argmax of a drifted Wiener process.

Weinstock inequality in higher dimensions

We prove that the Weinstock inequality for the first nonzero Steklov eigenvalue holds in $\mathbb{R}^n$, for $n \geq 3$, in the class of convex sets with prescribed surface area. The key result is a sharp isoperimetric inequality involving simultaneously the surface area, the volume and the boundary momentum of convex sets. As a by-product, we also obtain some isoperimetric inequalities for the first Wentzell eigenvalue.

Ambiguous reconstructions of hv-convex polyominoes

The core problem of discrete tomography, i.e., the faithful reconstruction of an unknown discrete object from projections along a given set of directions, is ill-posed in general. When further constraints are imposed on the object or on the employed directions, uniqueness of the reconstruction can be obtained. It is the case, for example, of convex lattice sets in Z2, for which a theorem by Gardner and Gritzmann assures the faithful reconstruction when suitable sets of directions are considered. It was conjectured that a similar result holds for the class of hv-convex polyominoes. In this paper we are concerned with this conjecture, providing new 4-tuples of discrete directions that do not lead to a unique reconstruction of hv-convex polyominoes, underlining the relevant structural difference with the class of convex sets. Our result is based on the recursive definition of new hv-convex switching components on discrete sets along four directions.

Cube Root Weak Convergence of Empirical Estimators of a Density Level Set

Given n independent random vectors with common density f on Rd, we study the weak convergence of three empirical-measure based estimators of the convex λ-level set Lλ of f, namely the excess mass set, the minimum volume set and the maximum probability set, all selected from a class of convex sets A that contains Lλ. Since these set-valued estimators approach Lλ, even the formulation of their weak convergence is non-standard. We identify the joint limiting distribution of the symmetric difference of Lλ and each of the three estimators, at rate n−1/3. It turns out that the minimum volume set and the maximum probability set estimators are asymptotically indistinguishable, whereas the excess mass set estimator exhibits “richer” limit behavior. Arguments rely on the boundary local empirical process, its cylinder representation, dimension-free concentration around the boundary of Lλ, and the set-valued argmax of a drifted Wiener process.

Existence of Fixed Points in a Class of Convex Sets

There are many well-known results about existence of fixed points for nonexpansive mappings defined on convex sets of Banach spaces which are \tau -compact, where \tau is usually a topology coarser than the norm topology. In this paper, we prove that it is possible to remove the \tau -compactness assumption and still obtain some fixed point results in a large framework. As particular cases, we can consider certain families of non- w^* -compact subsets in \ell_1 , the James space J_p , the trace space \mathcal{C}_1(\ell_2) and some other dual nonreflexive Banach spaces and we prove that „local" stability results for nonexpansive mappings can be achieved. In order to reach our purposes, we will make use of two new geometric coefficients a_\tau(X) and t_\tau(C) . Since the problem of the existence of fixed points for a nonexpansive mapping can equivalently be redefined assuming that the mapping is asymptotically regular, we will first focus on asymptotically regular mappings which satisfy a rather weak Lipschitz condition according to the value of the fraction \frac{a_\tau(X)}{t_\tau(C)} . Furthermore, we will show some examples proving that our results are sharp and can be used to obtain „global" stability results for the fixed point property.

Equicontinuous sweeping processes

We prove that the sweeping process on a regular class of convex sets is equicontinuous. Classes of polyhedral sets with a given finite set of normal vectors are regular, as well as classes of uniformly strictly convex sets. Regularity is invariant to certain operations on classes of convex sets such as intersection, finite union, arithmetic sum and affine transformation.

Further results on the classes of convex functions

This paper presents more results on the classes of convex sets and functions. The results are follow-up to similar results obtained on the classes of convex sets and functions by [1] and [9]. These relationships were further shown among convex sets which were not considered in the above references and more results were proposed and proved. JONAMP Vol. 11 2007: pp. 515 - 518

Laplacian eigenvalues for mean zero functions with constant Dirichlet data

We investigate the eigenvalues of the Laplace operator in the space of functions of mean zero and having a constant (unprescribed) boundary value. The first eigenvalue of such problem lies between the first two eigenvalues of the Laplacian with homogeneous Dirichlet boundary conditions and satisfies an isoperimetric inequality: in the class of open bounded sets of prescribed measure, it becomes minimal for the union of two disjoint balls having the same radius. Existence of an optimal domain in the class of convex sets is also discussed.

On a Class of Convex Sets and Functions

Given a convex subset C of ℝn, the set-valued mapping α↦α⋅C (where 0⋅C is, by convention, the recession cone of C) is increasing on ℝ+ if and only if C contains the origin, and decreasing on ℝ+ if and only if C is contained in its recession cone. This simple fact enables us to define a binary operation which combines a concave or convex function on ℝm with a convex subset of ℝn to produce a convex subset of ℝn+m. This binary operation is the set theoretic counterpart of a functional operation introduced by the author. In this paper, we present a detailed study of the class of convex subsets which are contained in their recession cones, and we establish some remarkable properties of our binary operation.

Computing an Integer Point of a Class of Convex Sets

A simplicial algorithm is proposed for computing an integer point of a convex set C⊂Rn satisfying

Denting point in the space of operator-valued continuous map

Denting point in the space of operator-valued continuous map

Singular sets of convex bodies and surfaces with generalized curvatures

In this paper we consider generalized surfaces with curvature measures and we study the properties of those k-dimensional subsets Σk of such surfaces where the curvatures have positive density with respect to k-dimensional Hausdorff measure. Special attention is given to boundaries of convex bodies inR3. We introduce a class of convex sets whose curvatures live only on integer dimension sets. For such convex sets we consider integral functionals depending on the curvature and the area ofK and on the curvature andHk of Σk.

The Gradient Projection Method Using Curry’s Steplength

It is shown that, using Curry’s steplength, the gradient projection method for finding constrained stationary points of a real valued $C^1 $ function on a closed convex subset C of Hilbert space can work for two rather different classes of convex sets: those with $C^2 $ boundary and “orthogonal polyhedra”. Both results are applications of Theorem 1, whose hypotheses require that the metric projection onto C possess directional derivatives which are continuous in a rather weak sense.

On Finite Classes of Convex Sets that Permit Space Coverings

On Finite Classes of Convex Sets that Permit Space Coverings

On Finite Classes of Convex Sets that Permit Space Coverings

On Finite Classes of Convex Sets that Permit Space Coverings

Glivenko–Cantelli theorems for classes of convex sets

Let denote the empirical distribution obtained from a sequence of i.i.d. -valued random vectors with common distribution P. If is a class of Borel subsets of then we say that it forms a Glivenko–Cantelli class for P if In this paper we describe a simple technique for identifying such classes, based on the idea of uniformity classes for setwise convergence. Classes for which the method proves successful include the closed half-spaces, closed balls, and the class of all convex subsets of .

Glivenko–Cantelli theorems for classes of convex sets

Let denote the empirical distribution obtained from a sequence of i.i.d. -valued random vectors with common distribution P. If is a class of Borel subsets of then we say that it forms a Glivenko–Cantelli class for P if In this paper we describe a simple technique for identifying such classes, based on the idea of uniformity classes for setwise convergence. Classes for which the method proves successful include the closed half-spaces, closed balls, and the class of all convex subsets of .

On the extension of additive functionals on classes of convex sets

Let Sf be a class of sets and let U(&) denote the class of all finite unions of sets from £f. This paper concerns itself with the question whether a vector valued function on Sf has an additive extension to U(£f). Several characterizations of functions with such an extension property are presented. One of these characterizations establishes a relationship between such extensions and integrals of certain types of simple functions. The special cases when Sf is the class of all convex polytopes or the class of all compact convex subsets of a euclidean space are investigated in more detail. Some examples are given to show that various extension problems that have been solved previously by methods particularly designed for each individual problem can also be solved by the application of these general results.

On classes of convex sets that permit plane coverings

A family {Ki¦ of convex domains in the Euclidean planeR2 is said to permit a plane covering if there exist rigid motions {τi} such that Ui∞=1τiKi =R2. Necessary and sufficient conditions that a given family of convex domains permits a plane covering are established.