Open Half-spaces Research Articles

On homotopy properties of solutions of some differential inclusions in the $W^{1,p}$-topology

We consider a differential inclusion on a manifold, defined by a field of open half-spaces whose boundary in each tangent space is the kernel of a one-form $\omega$. We make the assumption that the corank one distribution associated to the kernel of $\omega $ is completely nonholonomic of step $2$. We identify a subset of solutions of the differential inclusion, satisfying two endpoints and periodic boundary conditions, which are homotopically equivalent in the $W^{1,p}$-topology, for any $p\in [1,+\infty)$, to the based loop space and the free loop space respectively. We consider a differential inclusion on a manifold, defined by a field of open half-spaces whose boundary in each tangent space is the kernel of a one-form $\omega$. We make the assumption that the corank one distribution associated to the kernel of $\omega $ is completely nonholonomic. We identify a subset of solutions of the differential inclusion, satisfying two endpoints and periodic boundary conditions, which are homotopically equivalent in the $W^{1,p}$-topology, for any $p\in [1,+\infty)$, to the based loop space and the free loop space respectively.

Set-Valued Evenly Convex Functions: Characterizations and C-Conjugacy

In this work we deal with set-valued functions with values in the power set of a separated locally convex space where a nontrivial pointed convex cone induces a partial order relation. A set-valued function is evenly convex if its epigraph is an evenly convex set, i.e., it is the intersection of an arbitrary family of open half-spaces. In this paper we characterize evenly convex set-valued functions as the pointwise supremum of its set-valued e-affine minorants. Moreover, a suitable conjugation pattern will be developed for these functions, as well as the counterpart of the biconjugation Fenchel-Moreau theorem.

Some rigidity results for minimal graphs over unbounded Euclidean domains

<p style='text-indent:20px;'>We prove some new rigidity results for minimal graphs over unbounded Euclidean domains. In particular we prove that a positive minimal graph over an open affine half-space, and under the homogenous Dirichlet boundary condition, must be an affine function.</p>

Determination of the time-dependent change in the vapor concentration of 2,4,6-trinitrotoluene during the sublimation of its traces from the glass surface

The results of the measurements of 2,4,6-trinitrotoluene (TNT) vapor concentration over its trace amounts, called thin films, on the glass surface with a concentration of 100 ng/cm2 in a square area with a side of 1 cm over time are presented. The trace amounts of TNT on the glass were formed by applying a solution of TNT in the acetonitrile diluted with the chemically pure acetone, followed by the evaporation of the solvents. In order to measure the TNT vapor concentration, an EKHO-V-IDTS portable multibacillary gas-chromatograph with preliminary TNT vapor concentration was used. A sampling of the TNT vapor above the object was carried out with a remote vortex sampler. The vapor sample was taken from a distance of 2 cm from the glass surface. The concentration in the mode of the complete capture of TNT vapors was carried out to the stainless-steel wire mesh. The vapor concentration was determined from the chromatographic peak amplitude. It was found that the concentration of vapor over the examined surface with an area of 1 cm2 decreases from 10-13 to 10-14 g/cm3 within 2.6 ± 0.3 hours. TNT vapor concentration value of 10-14 g/cm3 corresponds to the threshold concentration of TNT vapor for the modern detectors. Based on the assumption that the vapor concentration is proportional to the amount of the TNT mass on the surface for the considered trace amounts of TNT, it was estimated that the initial surface concentration of trinitrotoluene of 100 ng/cm2 on the glass surface decreases to 12 ng/cm2 within 2.6 ± 0.3 hours due to sublimation into an open half-space. It was shown that the use of vortex sampling of vapor intensifies the sublimation of TNT from the glass surface.

The Double Queen Dido’s Problem

This paper deals with a variation of the classical isoperimetric problem in dimension $N\ge 2$ for a two-phase piecewise constant density whose discontinuity interface is a given hyperplane. We introduce a weighted perimeter functional with three different weights, one for the hyperplane and one for each of the two open half-spaces in which $\mathbb{R}^N$ gets partitioned. We then consider the problem of characterizing the sets $\Omega$ that minimize this weighted perimeter functional under the additional constraint that the volumes of the portions of $\Omega$ in the two half-spaces are given. It is shown that the problem admits two kinds of minimizers, which will be called type I and type II, respectively. These minimizers are made of the union of two spherical domes whose angle of incidence satisfies some kind of \textquotedblleft Snell's law\textquotedblright. Finally, we provide a complete classification of the minimizers depending on the various parameters of the problem.

Characterizations of Generalized Proximinal Subspaces in Real Banach Spaces

Let X be a real Banach space, C a closed bounded convex subset of X with the origin as an interior point, and $$p_C$$ the Minkowski functional generated by the set C. This paper is concerned with the problem of generalized best approximation with respect to $$p_C$$ . A property $$(\varepsilon _*)$$ concerning a subspace of $$X^*$$ is introduced to characterize generalized proximinal subspaces in X. A set C with feature as above in the space $$l_1$$ of absolutely summable sequences of real numbers and a continuous linear functional f on $$l_1$$ are constructed to show that each point in an open half space determined by the kernel of f admits a generalized best approximation from the kernel but each point in the other open half space does not.

Evenly convex credal sets

An evenly convex credal set is a set of probability measures that is evenly convex; that is, a set that is an arbitrary intersection of open halfspaces. An evenly convex credal set can for instance encode preference judgments through strict and non-strict inequalities such as P(A)>1/2 and P(A)≤2/3. This paper presents an axiomatization of evenly convex sets from preferences, where we introduce a new (and very weak) Archimedean condition. We examine the duality between preference orderings and credal sets; we also consider assessments of almost preference and natural extensions. We then discuss regular conditioning, a concept that is closely related to evenly convex sets.

The hamburger theorem

We generalize the ham sandwich theorem to d+1 measures on Rd as follows. Let μ1,μ2,…,μd+1 be absolutely continuous finite Borel measures on Rd. Let ωi=μi(Rd) for i∈[d+1], ω=min⁡{ωi;i∈[d+1]} and assume that ∑j=1d+1ωj=1. Assume that ωi≤1/d for every i∈[d+1]. Then there exists a hyperplane h such that each open halfspace H defined by h satisfies μi(H)≤(∑j=1d+1μj(H))/d for every i∈[d+1] and ∑j=1d+1μj(H)≥min⁡{1/2,1−dω}≥1/(d+1). As a consequence we obtain that every (d+1)-colored set of nd points in Rd such that no color is used for more than n points can be partitioned into n disjoint rainbow (d−1)-dimensional simplices.

Cutting convex curves

We show that for any two convex curves C1 and C2 in Rd parametrized by [0,1] with opposite orientations, there exists a hyperplane H with the following property: For any t∈[0,1] the points C1(t) and C2(t) are never in the same open half space bounded by H. This will be deduced from a more general result on equipartitions of ordered point sets by hyperplanes.

Lagrange Duality for Evenly Convex Optimization Problems

An evenly convex function on a locally convex space is an extended real-valued function, whose epigraph is the intersection of a family of open halfspaces. In this paper, we consider an infinite-dimensional optimization problem, for which both objective function and constraints are evenly convex, and we recover the classical Lagrange dual problem for it, via perturbational approach. The aim of the paper was to establish regularity conditions for strong duality between both problems, formulated in terms of even convexity.

Density of a semigroup in a Banach space

We study conditions on a set in a Banach space which are necessary or sufficient for the set of all sums , , to be dense in . We distinguish conditions under which the closure is an additive subgroup of , and conditions under which this additive subgroup is dense in . In particular, we prove that if is a closed rectifiable curve in a uniformly convex and uniformly smooth Banach space , and does not lie in a closed half-space , , and is minimal in the sense that every proper subarc of lies in an open half-space , then . We apply our results to questions of approximation in various function spaces.

Crooked halfspaces

We develop the Lorentzian geometry of a crooked halfspace in (2+1)-dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving an explicit slice for the action of the automorphism group. The set of parallelism classes of timelike lines, or particles , in a crooked halfspace is a geodesic halfplane in the hyperbolic plane. Every point in an open crooked halfspace lies on a particle. The correspondence between crooked halfspaces and halfplanes in hyperbolic 2-space preserves the partial order defined by inclusion, and the involution defined by complementarity. We find conditions for when a particle lies completely in a crooked half space. We revisit the disjointness criterion for crooked planes developed by Drumm and Goldman in terms of the semigroup of translations preserving a crooked halfspace. These ideas are then applied to describe foliations of Minkowski space by crooked planes.

Isoperimetric and Stable Sets for Log-ConcavePerturbations of Gaussian Measures

AbstractLet be an open half-space or slab in ℝn+1endowed with a perturbation of the Gaussian measureof the form f (p) := exp(ω(p) − c|p|2), where c > 0 and ω is a smooth concave function depending only onthe signed distance from the linear hyperplane parallel to ∂ Ω. In this work we follow a variational approach to show that half-spaces perpendicular to ∂ Ω uniquely minimize the weighted perimeter in Ω among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spacesparallel or perpendicular to ∂ Ω as the unique stable sets with small singular set and null weighted capacity.Our methods also apply for = ℝn+1, which produces in particular the classification of stable sets in Gaussspace and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to studythe weighted minimizers when the perturbation term ω is concave and possibly non-smooth.

Conditionally evenly convex sets and evenly quasi-convex maps

Evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces. We introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem. This notion is then applied to obtain the dual representation of conditionally evenly quasi-convex maps, which turns out to be a fundamental ingredient in the study of quasi-convex dynamic risk measures.

Duality for Closed Convex Functions and Evenly Convex Functions

We introduce two Moreau conjugacies for extended real-valued functions h on a separated locally convex space. In the first scheme, the biconjugate of h coincides with its closed convex hull, whereas, for the second scheme, the biconjugate of h is the evenly convex hull of h. In both cases, the biconjugate coincides with the supremum of the minorants of h that are either continuous affine or closed (respectively, open) halfspaces valley functions.

On minimal generators for semi-closed polyhedra

A semi-closed polyhedron is defined as the intersection of finitely many closed and/or open half-spaces, which can be regarded as an extension of a polyhedron. The same as a polyhedron, a semi-closed polyhedron admits both -representation and -representation. In this paper, we introduce the concept of a minimal -representation for semi-closed polyhedra which can be regarded as a natural extension of a minimal -representation for polyhedra. We derive some criteria for refining generators of a semi-closed polyhedron. These refining criteria can be applied to obtain a minimal generator for a semi-closed polyhedron.

Semi-monotone sets

A coordinate cone in \mathbb R^n is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is an open bounded subset of \mathbb R^n , definable in an o-minimal structure over the reals, such that its intersection with any translation of any coordinate cone is connected. This notion can be viewed as a generalization of convexity. Semi-monotone sets have a number of interesting geometric and combinatorial properties. The main result of the paper is that every semi-monotone set is a topological regular cell.

In This Issue

In This Issue

Null controllability of planar bimodal piecewise linear systems

This article investigates the null controllability of planar bimodal piecewise linear systems, which consist of two second order LTI systems separated by a line crossing through the origin. It is interesting to note that even when both subsystems are controllable in the classical sense, the whole piecewise linear system may be not null controllable. On the other hand, a piecewise linear system could be null controllable even when it has uncontrollable subsystems. First, the evolution directions from any non-origin state are studied from the geometric point of view, and it turns out that the directions usually span an open half space. Then, we derive an explicit and easily verifiable necessary and sufficient condition for a planar bimodal piecewise linear system to be null controllable. Finally, the article concludes with several numerical examples and discussions on the results and future work.

Binary Plane Partitions for Disjoint Line Segments

A binary space partition (BSP) for a set of disjoint objects in Euclidean space is a recursive decomposition where each step partitions the space (and possibly some of the objects) along a hyperplane and recurses on the objects clipped in each of the two open half-spaces. The size of a BSP is defined as the number of resulting fragments of the input objects. It is shown that every set of n disjoint line segments in the plane admits a BSP of size O(nlog n/log log n). This bound is the best possible.