Representation Of Convex Sets Research Articles

Dual representation of convex sets of probability measures on totally bounded spaces

Convex sets of probability measures, frequently encountered in probability theory and statistics, can be transparently analyzed by means of dual representations in a function space. This paper introduces totally bounded spaces, whose structure is defined by a set of bounded real-valued functions, as a general framework for studying such representations. The reinterpretation of classical theorems in this framework clarifies the role of compactness and leads to simple existence criteria. Applications include results on the existence of probability measures satisfying given sets of conditions and an equivalence of consistent preferences and families of probability measures. Moreover, countable additivity of probabilities is seen to be a consequence of elementary consistency assumptions.

Reachability analysis of linear systems using support functions

This work is concerned with the algorithmic reachability analysis of continuous-time linear systems with constrained initial states and inputs. We propose an approach for computing an over-approximation of the set of states reachable on a bounded time interval. The main contribution over previous works is that it allows us to consider systems whose sets of initial states and inputs are given by arbitrary compact convex sets represented by their support functions. We actually compute two over-approximations of the reachable set. The first one is given by the union of convex sets with computable support functions. As the representation of convex sets by their support function is not suitable for some tasks, we derive from this first over-approximation a second one given by the union of polyhedrons. The overall computational complexity of our approach is comparable to the complexity of the most competitive available specialized algorithms for reachability analysis of linear systems using zonotopes or ellipsoids. The effectiveness of our approach is demonstrated on several examples.

Approximation of a Metric on a Dominating Cone

A theorem on the representation of a pseudoball of a dominating cone is proven. This theorem is an analog of the Caratheodory theorem on the representation of convex sets. The theorem allows to develop numerical methods for quantitative evaluation of systems in many indicators.

Asymptotic geometry of high-density smooth-grained Boolean models in bounded domains

The purpose of the paper is to study the asymptotic geometry of a smooth-grained Boolean model (X[t])t≥0 restricted to a bounded domain as the intensity parameter t goes to ∞. Our approach is based on investigating the asymptotic properties as t → ∞ of the random sets X[t;β], β≥0, defined as the Gibbsian modifications of X[t] with the Hamiltonian given by βtμ(·), where μ is a certain normalized measure on the setting space. We show that our model exhibits a phase transition at a certain critical value of the inverse temperature β and we prove that at higher temperatures the behaviour of X[t;β] is qualitatively very similar to that of X[t] but it becomes essentially different in the low-temperature region. From these facts we derive information about the asymptotic properties of the original process X[t]. The results obtained include large- and moderate-deviation principles. We conclude the paper with an example application of our methods to analyse the asymptotic moderate-deviation properties of convex hulls of large uniform samples on a multidimensional ball. To translate the above problem to the Boolean model setting considered we use an appropriate representation of convex sets in terms of their support functions.

Asymptotic geometry of high-density smooth-grained Boolean models in bounded domains

The purpose of the paper is to study the asymptotic geometry of a smooth-grained Boolean model (X [t]) t≥0 restricted to a bounded domain as the intensity parameter t goes to ∞. Our approach is based on investigating the asymptotic properties as t → ∞ of the random sets X [t;β], β≥0, defined as the Gibbsian modifications of X [t] with the Hamiltonian given by βtμ(·), where μ is a certain normalized measure on the setting space. We show that our model exhibits a phase transition at a certain critical value of the inverse temperature β and we prove that at higher temperatures the behaviour of X [t;β] is qualitatively very similar to that of X [t] but it becomes essentially different in the low-temperature region. From these facts we derive information about the asymptotic properties of the original process X [t]. The results obtained include large- and moderate-deviation principles. We conclude the paper with an example application of our methods to analyse the asymptotic moderate-deviation properties of convex hulls of large uniform samples on a multidimensional ball. To translate the above problem to the Boolean model setting considered we use an appropriate representation of convex sets in terms of their support functions.

Dual Representations of Convex Sets and Gâteaux Differentiability Spaces

The duality of two kinds of representations of convex sets is studied: the tangential representation of a convex body and the representations of its polar or negative polar by means of their weak* exposed points. The equivalence of the representations is proved and a condition for their validity is obtained for individual sets (the case of arbitrary locally convex space) and for classes of sets (the case of Gâteaux differentiability locally convex space). Properties of Gâteaux differentiability locally convex spaces are studied and some examples of such spaces are given.

Ergodic decomposition of probability laws

The main results of the paper concern integral representations of convex sets of probability laws. Various types of simplices are introduced and characterized. Invariance with respect to Markovian operators plays a key role.