Closed convex set

Theorem 1.10. A closed convex set of a reflexive Banach space is weakly closed. 

Theorem 1.11. Let V be a reflexive Banach space, and K c V be a closed convex set. Assume that J V R is a coercive and weakly lower semicontinuous functional. Then the problem... 

Proof. To prove the existence of a solution, we implement the idea that was earlier used in a simpler case by (Shi, Shillor, 1992). We introduce two closed convex sets... 

Obviously, for (3.113) to hold, it suffices to choose 5 from the condition We define the bounded closed convex set... 

Imprecise probabilities Any of several theories involving models of uncertainty that do not assume a unique underlying probability distribution, but instead correspond to a set of probability distributions. An imprecise probability arises when one s lower probability for an event is strictly smaller than one s upper probability for the same event. Theories of imprecise probabilities are often expressed in terms of a lower probability measure giving the lower probability for every possible event from some universal set, or in terms of closed convex sets of probability distributions (which are generally much more complicated structures than either probability boxes or Dempster-Shafer structures). 

Let S be a nonempty closed convex set in 5Rn, and a vector y which does not belong to the set S. Then there exist a nonzero vector c and a scalar z such that... 

In a closed convex set 5 in I , there exists a unique point y, which is closest to a given point x outside the set S. 

Straight line sections arise in AR theory due to mixing. Feinberg and Hildebrandt (1997) provide a precise definition of these mixing lines specifically for use on the boundary of a closed convex set, which are termed lineations. An w-dimensional lineation is a set of points, L, on the boundary of convex set X with the following properties ... 

This assumption does not restrict the generality. We introduce the closed and convex set... 

We proceed with an investigation of the contact problem for a plate under creep conditions. We know that for every fixed / G L Q) there exists a unique solution w,M satisfying (2.35)-(2.37). Let G L Q) be a given element and F c (Q) be a closed convex and bounded set. We introduce the cost functional... 

Many methods in chemistry for the correlation energy are based on a form of perturbation theory, but the positivity conditions are quite different. Traditional perturbation theory performs accurately for all kinds of two-particle reduced Hamiltonians, which are close enough to a mean-field (Hartree-Fock) reference. There are a myriad of chemical systems, however, where the correlated wave-function (or 2-RDM) is not sufficiently close to a statistical mean field. Different from perturbation theory, the positivity conditions function by increasing the number of extreme two-particle Hamiltonians in which are employed as constraints upon the 2-RDM in Eq. (50) and, hence, they exactly treat a certain convex set of reduced Hamiltonians to all orders of perturbation theory. For the... 

Remark 1 A convex set may have no vertices (e.g., a line, an open ball), a finite number of vertices (e.g., a polygon), or an infinite number of vertices (e.g., all points on a closed ball). 

Show that an open and closed ball around a point 6 9 n is a convex set. 

An important property of the systems having a convex finite co-invariant set is as follows. Any closed convex finite co-invariant set has a steady-state point. This follows from the known Brower fixed-point theorem (see, for example, ref. 21), that has been extensively used in various fields of mathematics to prove theorems concerning the existence of solutions. 

In Chapter 3, we returned to the BTX beaker experiment and used the ideas of mixing and convex hulls from Chapter 2 to improve the maximum concentration of toluene. This chapter also introduced the idea of a candidate AR specific to the system of interest. The AR for the BTX system was revealed as the limiting case of infinitely many batch mixing and reaction experiments, which were conducted in a serial fashion. Chapter 3 also introduced a number of necessary conditions of the AR. Since mixing is a linear process, the AR must be a closed, compact, and convex set of points in concentration space that is formed by the convex hull of all achievable points. 

The complement principle uses the idea of closures of a convex set X, conv(X). The closure of con(X), cl conv(X), is the smallest closed subset of conv(X). The closure of conv(X) is used to represent the set of all points in conv(X) including any points that might only be obtained in the limit of a process. cl conv(X) is used to include points in conv(X) that might not be physically achievable (i.e., equilibrium points). 

A hyperplane H is a supporting hyperplane of a convex set B if B is entirely contained in one of the two closed half-spaces determined by H and B has at least one boundary-point on H. 

Let K c V he a convex closed set. We assume that y is a strictly convex reflexive Banach space. For given u G V an element Pu G K is called a projection of u onto the set K if... 

Let K cV he a. convex closed subset of a reflexive Banach space V, I he a duality mapping, and P be a projection operator of V onto K. We are in a position to give a definition of a penalty operator. An operator (5 V V is called a penalty operator connected with the set K if the following conditions are fulfilled. Firstly, / is a monotonous bounded semicontinuous operator. Secondly, a kernel of / coincides with K, i.e. 

This result enables us to investigate the extreme crack shape problem. The formulation of the last one is as follows. Let C Hq 0, 1) be a convex, closed and bounded set. Assume that for every -0 G the graph y = %j) x) describes the crack shape. Consequently, for a given -0 G there exists a unique solution of the problem... 

In the sequel we shall study an optimal control problem. Let C (fl) be a convex, bounded and closed set. Assume that ( < 0 on T for each G. In particular, this condition provides nonemptiness for Kf. Denote the solution of (2.131) by % = introduce the cost functional... 

Consider the problem of finding the extreme crack shapes. The setting of this problem is as follows. Let C be a convex, closed and... 

Let C (G) be a convex, closed and bounded set such that every element -0 6 satisfies the following relations ... 

Does the following set of constraints that form a closed region form a convex region ... 

Solution. See Figure E4.9 for the region delineated by the inequality constraints. By visual inspection, the region is convex. This set of linear inequality constraints forms a convex region because all the constraints are concave. In this case the convex region is closed. 

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