Metric space of unlimited convex sets and unlimited polyhedron

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Астраханский государственный архитектурно - строительный университет

METRIC SPACE OF UNLIMITED CONVEX SETS AND UNLIMITED POLYHEDRON

Аннотация

МЕТРИЧЕСКОЕ ПРОСТРАНСТВО НЕОГРАНИЧЕННЫХ ВЫПУКЛЫХ МНОЖЕСТВ И НЕОГРАНИЧЕННЫЕ МНОГОГРАННИКИ

В работе дается определение метрического пространства H(K) неограниченных замкнутых выпуклых подмножеств банахового пространства X, имеющих один и тот же рецессивный K. В качестве расстояния используется метрика Хаусдорфа. В настоящей работе установлено, что свойства H(K) метрического пространства отличаются от свойств метрического пространства выпуклых компактов с метрикой Хаусдорфа. Установлено, что теорема аналогичная теореме об аппроксимации выпуклых компактов многогранниками неверна. То есть не каждый элемент метрического пространства H(K) может быть аппроксимирован обобщенными многогранниками, являющихся аналогами обычных многогранников. В работе вводится понятие обобщенного многогранника следующим образом. Элементы совокупности H(0)+K называются обобщенным многогранниками.

Выведен критерий аппроксимации. Для того, чтобы элемент пространства H(K) мог быть аппроксимирован обобщенными многогранниками в метрике Хаусдорфа необходимо и достаточно, чтобы его опорная функция была равномерно непрерывной.

Ключевые слова: Пространство неограниченных замкнутых выпуклых множеств, метрика Хаусдорфа, опорная функция, нормальный конус, рецессивный конус.

Abstract

In the paper there is a definition of metric space H(K) the unlimited closed convex subsets of Banach space X, having the same recessive K. There is a Hausdorff metric that is used as a distance. It is established in this paper that the properties of a metric space H(K) are different from the properties of the metric spaceof convex compacts with Hausdorff metric. It is established that the theorem similar to the theorem of approximation of convex compacts polyhedrons is wrong. That is not each element of metric space H(K) can be approximated by the generalized polyhedrons, which are the analogues of the normalpolyhedrons.

The paper introduces the concept of a generalized polyhedron in the following way. The set of elements H(0)+K are known as generalized polyhedrons.

The criterion of approximation is derived. In order for the element of the space H(K) could be approximated by generalized polyhedrons in the Hausdorff metric it is necessary and sufficient that its basic function was evenly continuous.

Keywords: a space of unlimited closed convex sets, Hausdorff metric, support function, normal cone, recessive cone.

Main text

convex unlimited metric space

The paper examines the unlimited closed convex subsets of Banach space X, having the same recessive cone, and metric spaces, which they form with the Hausdorff metric. The aim of this work is the study of the properties of this metric space, as well as receiving an analog of the known theorem of approximation of convex compacts by normal polyhedrons.Convex sets and their properties were studied in the works [1-3]. The necessary information according to the functional analysis needed for the proof of the main theorem is taken from [1,2].

The space x is assumed to be reflexive, it is in the paper and, therefore, the limited, convex, closed sets are weakly compact in it. Suppose

Sets are closed single spheres from accordingly; - single sphere Y from and etc. Apparently . Denote by

the extended number line.

Definition 1. Cone, maximal by inclusion among cones satisfies the condition , is called the recessive cone of the set. You can check that it exists and that it is convex, closed, if the set is closed, convex.

Definition 2. The function on is called the supporting function of the set A. The function equal to zero on A, and to plus infinity outside A is called the indicator function of the set.

Definition 3. The epigraph of this function is called the set and denote by epif. A function is called closed when the epigraphis closed in the direct multiplication . Function is called its own if , .

By the G(f) we denote the graph of the function f.

Lemma 1. A function is closed then and only then when it is semi continuous from below [1].

Lemma 2. The following statements are true: a) the support function of an arbitrary set is always convex, homogeneous and closed in the weak topology, and therefore, because of the reflexivity of the space it is closed, and by Lemma 1 the support function is always semi continuous from below;

b) for any set A,B numbers it is true: ;

c) for convex closed sets is true: ;

d) suppose is a convex, homogeneous, semi continuous from below and it's own function, then there is a not empty closed convex set, which support function is .

Lemma 3. Suppose A,B is closed, convex subsets X. Let us define two numbers:

Then it is true that . This number is called the Hausdorff distance between the sets A and B.

The set of all non-empty, closed, convex subsets, having the same recessive cone K, remote from it at a finite Hausdorff distance is a metric space; let us denote it H(K). Thus, some closed, convex sets fall out of our consideration. Owing to reflexivity X elements from H(0) - is convex weakly compact and therefore

The aim of this work is to find out under what conditions , and when ? Denote by K* the cone associated to the cone K that is .

Lemma 4. Support function of any element is limited on the set

and is equal to plus infinity outside the cone . Moreover

Definition 4. The cone K is called normal if

Lemma 5. Suppose the cone K is normal, then there exists a normal physical cone K₁ such that K?K₁[2].

Lemma 6. Suppose X=X**; A,B is weakly closed subsets X. If the set A?B can be putted by a shift to a normal cone then the algebraic sum A+B is weakly closed [3].

Lemma 7. Suppose K is normal,

A,B?H(K); б,в?0, then бA+вB?H(K).

Definition 5. Suppose Щ an arbitrary subset of space X. Function is called uniform by the Hausdorff, if there is a nonzero area д, of the graph of a function f, not intersecting with the graph of the function

Lemma 8. Suppose the set Щ is convex set and the function f is limited, that is . Then the function f is uniform by the Hausdorff, then and only then, when it is uniformly continuous.

Lemma 9. Suppose Щ is convex, open subset, and f is convex, limited function to the Щ, , A - is closed, convex subset lying in Щ with some closed area of non-zero radius д that is . Then the function satisfies the Lipschitz condition on A with as constant. It is enough to prove the Lemma for one-dimensional case.

Lemma 10. Suppose the function f uniformly continuous on limited convex set Щ, then the function f is limited on it.

THEOREM 1. Suppose The element belongs to the circuit H(0) + K then and then only when the support function of the element A is uniformly continuous in the element in the area .

Let us find out in the finite-dimensional case: when the metric subspace H(0) + K densely in H(K)? Here and after .

Definition 6. The intersection of a finite number of closed half-spaces is called a polyhedron.

Definition 7. The point is called an extreme point of the set A, if from the multiple follows that .

Lemma 11. Suppose Щ is polyhedron from . Then every finite convex, semi continuous from below function is continuous on it.

Lemma 12. Suppose Щ is convex compact from and every limited, convex, semi continuous from below function is continuous on it, then the set Щ is always a polyhedron.

THEOREM 2. Considering space . Metric space H(0) + K is densely in H(K) that is then, and only then when con K is a polyhedron.

The RESULT (consequence). Closed, convex set A from is approximated by polyhedrons with arbitrary precision then, and only then, when recessive cone K of a set A is a polyhedron and .

In the two-dimensional case, the space of unbounded convex closed sets H(K) is simple. In two-dimensional space of any cone is unbounded polyhedron. Therefore, in this case any element of the space H(K) can be approximated by generalized polyhedra. But in the three-dimensional case not every element of the space of unbounded convex sets H(K) is approximated by generalized polyhedra.

Let us give an example of the set not belonging to the circuit H(0) + K. Suppose

It is easy to see that the recessive cone of the set A is K and . Let us show that the support function of the set is discontinuous at the point . We have and for any vector b* not proportional to the vector a* and taken from the border of the cone .

Remark. The algebraic sum , where is polyhedron from H(0) we will call a generalized polyhedron.

Since in the element A from H(0) always can be approximated by polyhedrons, then theorem 1 in finite-dimensional case will be: the element A from H(0) can be approximated with arbitrary precision by generalized polyhedrons then and only then, when its supporting function is continuous in the cone .

The results of the paper can be used in the theory of approximation of convex functions defined on all finite-dimensional space.

References

1. Иоффе, А.Д. Теория экстремальных задач / А.Д. Иоффе, В.М. Тихомиров. М.: Наука, 1974. 450 с.

2. Ажоркин, В.И. К геометрии конусов линейных положительных операторов в пространстве Банаха / В.И. Ажоркин, И.А. Бахтин // Труды центрального зонального объединения математических кафедр. Функциональный анализ и теория функций. 1971. Вып. 2. C. 3-10.

3. 3.Яксубаев, К.Д. О замкнутости алгебраической суммы замкнутых множеств / К.Д. Яксубаев // Доклады академии наук Уз ССР. 1984. № 10. C. 10-11.

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