Convex Subset Of Euclidean Space Research Articles

Asymptotic base loci via Okounkov bodies

An Okounkov body is a convex subset of Euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag. In this paper, we recover the asymptotic base loci from the Okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving Seshadri constants. Consequently, we obtain the nefness and ampleness criteria of divisors in terms of the Okounkov bodies. Furthermore, we compute the divisorial Zariski decomposition by the Okounkov bodies, and find upper and lower bounds for moving Seshadri constants given by the size of simplexes contained in the Okounkov bodies.

Optimal Control of a Partially Observable Failing System with Costly Multivariate Observations

We model a partially observable deteriorating system subject to random failure. The state process follows an unobservable continuous time homogeneous Markov chain. At equidistant sampling times vector-valued observations having multivariate normal distribution with state-dependent mean and covariance matrix are obtained at a positive cost. At each sampling epoch a decision is made either to run the system until the next sampling epoch or to carry out full preventive maintenance, which is assumed to be less costly than corrective maintenance carried out upon system failure. The objective is to determine the optimal control policy that minimizes the long-run expected average cost per unit time. We show that the optimal preventive maintenance region is a convex subset of Euclidean space. We also analyze the practical three-state version of this problem in detail and show that in this case the optimal policy is a control limit policy. An efficient computational algorithm is developed for the three-state problem, illustrated by a numerical example.

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.

Hyperspaces of max-plus convex subsets of powers of the real line

The notion of a max-plus convex subset of Euclidean space can be naturally extended to other linear spaces. The aim of this paper is to describe the topology of hyperspaces of max-plus convex subsets in Tychonov powers Rτ of the real line. We show that the corresponding spaces are absolute retracts if and only if τ≤ω1.

Image Manifolds which are Isometric to Euclidean Space

Recently, the Isomap procedure [10] was proposed as a new way to recover a low-dimensional parametrization of data lying on a low-dimensional submanifold in high-dimensional space. The method assumes that the submanifold, viewed as a Riemannian submanifold of the ambient high-dimensional space, is isometric to a convex subset of Euclidean space. This naturally raises the question: what datasets can reasonably be modeled by this condition? In this paper, we consider a special kind of image data: families of images generated by articulation of one or several objects in a scene--for example, images of a black disk on a white background with center placed at a range of locations. The collection of all images in such an articulation family, as the parameters of the articulation vary, makes up an articulation manifold, a submanifold of L2. We study the properties of such articulation manifolds, in particular, their lack of differentiability when the images have edges. Under these conditions, we show that there exists a natural renormalization of geodesic distance which yields a well-defined metric. We exhibit a list of articulation models where the corresponding manifold equipped with this new metric is indeed isometric to a convex subset of Euclidean space. Examples include translations of a symmetric object, rotations of a closed set, articulations of a horizon, and expressions of a cartoon face. The theoretical predictions from our study are borne out by empirical experiments with published Isomap code. We also note that in the case where several components of the image articulate independently, isometry may fail; for example, with several disks in an image avoiding contact, the underlying Riemannian manifold is locally isometric to an open, connected, but not convex subset of Euclidean space. Such a situation matches the assumptions of our recently-proposed Hessian Eigenmaps procedure, but not the original Isomap procedure.

Strategy-Proof Probabilistic Mechanisms in Economies with Pure Public Goods

Public good economies where agents are endowed with strictly convex continuous single-peaked preferences on a convex subset of Euclidean space are considered. Such an economy arises for instance in the classical problem of allocating a given budget to finance the provision of several public goods where the agents have monotonically increasing strictly convex continuous preferences. A probabilistic mechanism assigns a probability distribution over the feasible alternatives to any profile of reported preferences. The main result of the paper establishes that any strategy-proof (in the sense of A. Gibbard, Econometrica 45 (1977), 665–681) and unanimous mechanism must be a random dictatorship. Journal of Economic Literature Classification Numbers: D70, D71, H40, C60.

Even valuations on convex bodies

The notion of even valuation is introduced as a natural generalization of volume on compact convex subsets of Euclidean space. A recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on Grassmannians. This connection can be described in part using generating distributions for symmetric compact convex sets. We also explore some consequences of these characterization results in convex and integral geometry.

On existence of positive periodic solutions

We present the geometric method for detecting periodic solutions of time periodic nonautonomous differential equations in interior of convex subset of euclidean space. The method is based on the Lefschetz fixed point theorem and the topological principle of Wazewski. Two applications to the existence of positive periodic solutions are considered.

The structure of continuous-valued neutral monotonic social functions

This article consists of several results characterizing neutral monotonic social functions over alternative sets that are compact, convex subsets of Euclidean space. One major result is that a neutral monotonic social function is continuous-valued (i.e., does not make abrupt reversals) for all profiles of continuous-valued weak orderings if and only if it is simple (i.e., completely determined by a single set of decisive coalitions). A second major result is that a continuous-valued neutral monotonic social function will guarantee the existence of a socially undominated alternative for all profiles of continuousvalued, almost convex weak orderings if and only if the smallest empty intersection within the set of decisive coalitions is at least m+2 in size where m is the dimension of the alternative set.

The convex hull of a spherically symmetric sample

Using the isomorphism between convex subsets of Euclidean space and continuous functions on the unit sphere we describe the probability measure of the convex hull of a random sample. When the sample is spherically symmetric the asymptotic behavior of this measure is determined. There are three distinct limit measures, each corresponding to one of the classical extreme-value distributions. Several properties of each limit are determined.

The convex hull of a spherically symmetric sample

Using the isomorphism between convex subsets of Euclidean space and continuous functions on the unit sphere we describe the probability measure of the convex hull of a random sample. When the sample is spherically symmetric the asymptotic behavior of this measure is determined. There are three distinct limit measures, each corresponding to one of the classical extreme-value distributions. Several properties of each limit are determined.