Convex lot and its properties. Define a convex set

In which all points of the segment formed by any two points of a given set also belong to this set.

Definitions

Examples

• Convex subsets of a set $\backslash R$(a bunch of real numbers) represent intervals from $\backslash R$.
• Examples of convex subsets in two-dimensional Euclidean space ( $\backslash R^2$) are regular polygons.
• Examples of convex subsets in three-dimensional Euclidean space ( $\backslash R^3$) are Archimedean solids and regular polyhedra.
• Keppler-Poinsot bodies (regular star-shaped polyhedra) are examples of non- convex sets.

Properties

• A convex set in a topological linear space is connected and path-connected, homotopy equivalent to a point.
• In terms of connectivity, a convex set can be defined as follows: a set is convex if its intersection with any (real) line is connected.
• Let $K$- a convex set in linear space. Then for any elements $u\_1,\backslash ;u\_2,\backslash ;\backslash ldots,\backslash ;u\_r$ owned $K$ and for all non-negative $\backslash lambda\_1,\backslash ;\backslash lambda\_2,\backslash ;\backslash ldots,\backslash ;\backslash lambda\_r$, such that $\backslash lambda\_1+\backslash lambda\_2+\backslash ldots+\backslash lambda\_r=1$, vector $w=\backslash sum\_(k=1)^r\backslash lambda\_k\; u\_k$

belongs $K$.

• Vector $w$ is called a convex combination of elements $u\_1,\backslash ;u\_2,\backslash ;\backslash ldots,\backslash ;u\_r$.

Variations and generalizations

• Without any changes, the definition works for affine spaces over an arbitrary extension of the field of real numbers.

see also

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Literature

• Polovinkin E. S., Balashov M. V. Elements of convex and strongly convex analysis. - M.: FIZMATLIT, 2004. - 416 p. - ISBN 5-9221-0499-3..
• Timorin V. A.. - M.: MTsNMO, 2002. - 16 p. - ISBN 5-94057-024-0..

Links

An excerpt characterizing a convex set

And Natasha stood on tiptoe and walked out of the room the way dancers do, but smiling the way only happy 15-year-old girls smile. Having met Sonya in the living room, Rostov blushed. He didn't know how to deal with her. Yesterday they kissed in the first minute of the joy of their date, but today they felt that it was impossible to do this; he felt that everyone, his mother and sisters, looked at him questioningly and expected from him how he would behave with her. He kissed her hand and called her you - Sonya. But their eyes, having met, said “you” to each other and kissed tenderly. With her gaze she asked him for forgiveness for the fact that at Natasha’s embassy she dared to remind him of his promise and thanked him for his love. With his gaze he thanked her for the offer of freedom and said that one way or another, he would never stop loving her, because it was impossible not to love her.
“How strange it is,” said Vera, choosing a general moment of silence, “that Sonya and Nikolenka now met like strangers.” – Vera’s remark was fair, like all her comments; but like most of her remarks, everyone felt awkward, and not only Sonya, Nikolai and Natasha, but also the old countess, who was afraid of this son’s love for Sonya, which could deprive him of a brilliant party, also blushed like a girl. Denisov, to Rostov’s surprise, in a new uniform, pomaded and perfumed, appeared in the living room as dandy as he was in battle, and as amiable with ladies and gentlemen as Rostov had never expected to see him.

Returning to Moscow from the army, Nikolai Rostov was received by his family as best son, hero and beloved Nikolushka; relatives - as a sweet, pleasant and respectful young man; acquaintances - like a handsome hussar lieutenant, a deft dancer and one of the best grooms in Moscow.
The Rostovs knew all of Moscow; this year the old count had enough money, because all his estates had been remortgaged, and therefore Nikolushka, having got his own trotter and the most fashionable leggings, special ones that no one else in Moscow had, and boots, the most fashionable, with the most pointed socks and little silver spurs, had a lot of fun. Rostov, returning home, experienced a pleasant feeling after some period of time trying on himself to the old living conditions. It seemed to him that he had matured and grown very much. Despair for failing to pass an exam according to the law of God, borrowing money from Gavrila for a cab driver, secret kisses with Sonya, he remembered all this as childishness, from which he was now immeasurably far away. Now he is a hussar lieutenant in a silver mentic, with a soldier's George, preparing his trotter to run, together with famous hunters, elderly, respectable. He knows a lady on the boulevard whom he goes to see in the evening. He conducted a mazurka at the Arkharovs’ ball, talked about the war with Field Marshal Kamensky, visited an English club, and was on friendly terms with a forty-year-old colonel whom Denisov introduced him to.
His passion for the sovereign weakened somewhat in Moscow, since during this time he did not see him. But he often talked about the sovereign, about his love for him, making it felt that he was not telling everything yet, that there was something else in his feelings for the sovereign that could not be understood by everyone; and with all my heart he shared the general feeling of adoration in Moscow at that time for Emperor Alexander Pavlovich, who in Moscow at that time was given the name of an angel in the flesh.
During this short stay of Rostov in Moscow, before leaving for the army, he did not become close, but on the contrary, broke up with Sonya. She was very pretty, sweet, and obviously passionately in love with him; but he was in that time of youth when there seems to be so much to do that there is no time to do it, and the young man is afraid to get involved - he values ​​​​his freedom, which he needs for many other things. When he thought about Sonya during this new stay in Moscow, he said to himself: Eh! there will be many more, many more of these, somewhere, still unknown to me. I’ll still have time to make love when I want, but now there’s no time. In addition, it seemed to him that there was something humiliating for his courage in female society. He went to balls and sororities, pretending that he was doing it against his will. Running, an English club, carousing with Denisov, a trip there - that was another matter: it was befitting of a fine hussar.
At the beginning of March, the old Count Ilya Andreich Rostov was preoccupied with arranging a dinner at an English club to receive Prince Bagration.
The Count in a dressing gown walked around the hall, giving orders to the club housekeeper and the famous Theoktistus, the senior cook. English club, about asparagus, fresh cucumbers, strawberries, veal and fish for Prince Bagration’s dinner. The Count, from the day the club was founded, was its member and foreman. He was entrusted by the club with arranging a celebration for Bagration, because rarely did anyone know how to organize a feast in such a grand manner, hospitably, especially because rarely did anyone know how and want to contribute their money if they were needed to organize the feast. The cook and housekeeper of the club listened to the count's orders with cheerful faces, because they knew that under no one else could they profit better from a dinner that cost several thousand.

A set X is called convex if for any two of its points A,B ∈ X all points of the segment also belong to the set X, that is, if for any two of its points A,B ∈ X and for any value α in the point M = αA + (1 − α)B also belongs to the set X: M ∈ X.

Let X1, ...Xn be given - convex sets. Let us denote Y =Xi - the intersection of convex sets. Let us show that Y is a convex set. To do this, we show that for any points A,B ∈ Y and for any value of α in the point M = αA + (1 − α)B also belongs to the set Y: M ∈ Y . Since Y is the intersection of convex sets X1, ...Xn, then chosen arbitrarily points A,B belong to each of these sets Xi, i = 1..n. Due to the convexity of each of the sets Xi, by definition it follows that for an arbitrarily chosen value α ∈ the point M = αA+(1−α)B belongs to each of the sets (all of them are convex and contain A,B). Since all sets Xi contain the point M, then

the intersection of these sets also contains a point M: M ∈ Y. From last inclusion into force randomness A,B∈ Y and the arbitrariness of the parameter α ∈ implies the convexity of the set Y , as required to be shown.

95. Is the set of points satisfying the condition convex? Justify your answer.

Yes, it is obvious that this equality defines a linear half-plane in R4.

Let's justify this by definition:

A = (a1, a2, a3, a4), B= (b1, b2, b3, b4) ∈ X,

satisfying the above inequality.

Let's consider arbitrary point M = αA + (1 − α)B, where α ∈ is an arbitrary parameter value. ThenM(m1,m2,m3,m4) = αA + (1 − α)B

m1 = αa1 + (1 − αb1)

m2 = αa2 + (1 − αb2)

m3 = αa3 + (1 − αb3)

m4 = αa4 + (1 − αb4)

satisfiability of the given inequality:

5 + 2m1 + 3m2 − m3 + 5m4 ≥ 0

5 + 2(αa1 + (1 − αb1)) + 3(αa2 + (1 − αb2)) − (αa3 + (1 − αb3)) + 5(αa4 + (1 − αb4)) ≥ 0

Let's imagine 5 = α5+(1−α)5, expand and group the terms for ai and bi. We get:

α(5 + 2a1 + 3a2 − a3 + 5a4) + (1 − α)(5 + 2b1 + 3b2 − b3 + 5b4) ≥ 0

Since points A, B lie in the set X, their coordinates satisfy the inequality

to the one who sets the set. This means that both terms are non-negative due to the non-negativity

α and 1 − α. Therefore, the last inequality holds for any A, B and any value

parameter α ∈ . By definition, we have shown that a given set X is

convex.

96. Is the set of points satisfying the condition convex? Justify your answer.

Yes, it is obvious that this equality defines a linear hyperplane in R4.

Let's justify this by definition:

Consider any two points of this space

A = (a1, a2, a3, a4), B= (b1, b2, b3, b4) ∈ X

satisfying the above equality.

Consider an arbitrary point M = αA + (1 − α)B, where α ∈ is an arbitrary value of the parameter. Then M(m1,m2,m3,m4) = αA + (1 − α)B

m1 = αa1 + (1 − αb1)

m2 = αa2 + (1 − αb2)

m3 = αa3 + (1 − αb3)

m4 = αa4 + (1 − αb4)

Let us check for a point M(m1,m2,m3,m4) membership in the set X using

feasibility given equality:

m1 + 2m2 − 3m3 + 4m4 = 55

(αa1 + (1 − αb1)) + 2(αa2 + (1 − αb2)) − 3(αa3 + (1 − αb3)) + 4(αa4 + (1 − αb4)) = 55

Let's open the brackets and group the terms for ai and bi. We get:

α(a1 + 2a2 − 3a3 + 4a4) + (1 − α)(b1 + 2b2 − 3b3 + 4b4) = 55

Since points A, B lie in the set X, their coordinates satisfy the equality

defining the set, that is, (a1 + 2a2 − 3a3 + 4a4) = 55 and (b1 + 2b2 − 3b3 + 4b4) = 55.

Substituting these equalities into the last expression we get:

α55 + (1 − α)55 = 55

The last equality holds for any A,B and any value of the parameter α ∈ . By definition, we have shown that this set X is convex.

97. Give examples of a convex set: a) having a corner point; b) without a corner point. Can an unbounded convex set have a corner point? Give an example.

a) a square has 4 corner points

b) a circle has no corner points

c) an unlimited set can have corner points: it has one corner point (0;0)

98. Define the convex hull of a system of points. Let be the convex hull of the points , , , . Do the points: , ? Justify your answer.

that is, the condition is satisfied that this is a convex linear combination, which means X is part of the convex hull. Suppose that Y is also included in the convex combination, then all points of the segment must be included in the linear combination, but from the initial points it is clear (they are all located to the right of the straight line x = -1) that the entire convex combination is located to the right of the straight line x = -1, and point Y is on the left, which confirms that neither the entire segment nor point Y belong to the convex hull.

Convex set- a subset of Euclidean space containing a segment connecting any two points of this set.

Definition

In other words, a set is called convex if:

That is, if the set X together with any two points that belong to this set, contains a segment connecting them:

In space, convex sets are a straight line, a half-line, a segment, an interval, and a single-point set.

In space, the space itself, any of its linear subspaces, a ball, a segment, a one-point set, will be convex. Also, the following sets will be convex:

• hyperplane H p? with normal p :

• half-spaces into which hyperplanes divide space:

All of the listed sets (except for the bullet) are a special case of the convex set of polyhedra.

Properties of convex sets

• The intersection of convex sets is convex.
• A linear combination of points of a convex set is convex.
• A convex set contains any convex combination of its points.
• Any point n-dimensional Euclidean space with the convex hull of a set can be represented as a convex combination of at most n+1 points of this set

Let's consider n is a dimensional Euclidean space and let - a point in this space.

Consider two points and , belonging to .Set of points , which can be represented in the form

(in coordinates this is written like this:

segment, connecting the points and . The points themselves are called ends of the segment. In cases n=2 and n=3 is a segment in the usual sense of the word on a plane or in space (see Fig. 12). Note that for  =0, and for  =1, i.e. when  =0 and  =1 the ends of the segment are obtained.

Let in

given k points

. Dot

where everything is called convex combination dots

Let there be some region in space (in other words,

G there is some set of points from

).

Definition. The set (area) is called convex, if from what follows, that for   . In other words, G a convex set if it, together with any two of its points, contains a segment connecting these points.

In these figures, “a” and “b” are convex sets, and “c” is not a convex set, since it contains a pair of points such that the segment connecting them does not entirely belong to this set.

Theorem 1. Let G be a convex set. Then any convex combination of points belonging to this set also belongs to this set.

Proof

Let's prove the theorem using the method mathematical induction. At k=2 theorem is true, since it simply goes into the definition of a convex set.

Let the theorem be true for some k. Let's take a point and consider a convex combination

where is everyone

And .

Let's imagine

as

The theorem is proven.

Theorem 2. Admissible domain of the problem linear programming is a convex set.

Proof.

1. In standard form in matrix notation valid area G is determined by the condition

Those. x belongs to G and is therefore convex.

2. In canonical form area G is defined by the conditions

Let and belong to G, i.e.

.

those. and therefore G is convex. The theorem is proven.

Thus, the feasible region in a linear programming problem is a convex set. By analogy with two-dimensional or three-dimensional cases, for any n this area is called convex

polyhedron n- dimensional space

Theorem 3. The set of optimal plans for a linear programming problem is convex (if it is not empty).

Proof

If the solution to a linear programming problem is unique, then it is convex by definition - the point is considered a convex set. Let us now have two optimal plans for the linear programming problem.

those. there is also an optimal plan and, due to this, the set of optimal plans is convex. The theorem is proven.

Theorem 4. In order for a linear programming problem to have a solution, it is necessary and sufficient that objective function on the admissible set was bounded from above (when solving a maximum problem) or from below (when solving a minimum problem).

We give this theorem without proof.

Let X, at, z– elements n-dimensional real Euclidean space We will also call them vectors or points of space

Definition . A line segment connecting points x And y, is called a set of points of the form

Definition . The set of points is called convex set, if the segment connecting any two points is included in the set M, that is

For example, convex sets are a point, a segment, an open and closed parallelepiped space, an open and closed ball. Empty set is not convex.

Theorem . The non-empty intersection of any number of convex sets is a convex set.

Proof . Let be convex sets and points x, y belong to all these sets simultaneously; therefore, by the definition of a convex set, a Point belongs to all sets simultaneously. Thus, for any two points the points belong to the set M. Therefore, by definition M– convex set.

Definition . Hyperplane in is called the set of points

Where a–n-dimensional guide vector, parentheses indicate scalar product real number With is called a free member.

Notes . 1) A hyperplane is a convex set. Indeed, let Then for any point belongs G, because

2) Guide vector a orthogonal to the hyperplane, that is, for any vector z = x – y, connecting two arbitrary non-coinciding points of the hyperplane ( a, z) = 0. Indeed,

(a, z) = (a, x) – (a, y) = c – c = 0.

Definition . Many viewpoints

called half-space V

The direction of inequality in the definition can also be taken in the opposite direction.

Comment . A half-space is a convex set. Indeed, let Then for any point belongs S, because

Definition . Non-empty intersection finite number half-spaces are called convex polyhedron.

The use of the term convex polyhedron is explained by the fact that a half-space is a convex set, and a non-empty intersection of a finite number of convex sets is a convex set.

Definition . Many kinds

called positive orthant.

A positive orthant is a convex polyhedron. Indeed, inequality can be interpreted as a system of inequalities

Definition . Let a convex polyhedron G given by the system of inequalities

where are the direction vectors, k > n. If the point turns into equalities at least n inequalities, and the rank of the corresponding system of vectors is equal to n, then point at called angular(or extreme) point of the polyhedron.

Note that the number of corner points convex polyhedron maybe (depending on n And k) very large. Yes, when n = 10, k= 20 this number can be compared to 10 11.

Comment . Since equality of the form

can be replaced by a system of two inequalities

then if in the definition some of the inequalities (or all of the inequalities) are replaced by the corresponding equalities, then the resulting system of conditions also defines a convex polyhedron.

Let us recall the definition of a frequently used convex set.

Definition . ε – a neighborhood of a point is an open ball

Obviously, ε, a neighborhood of a point, is a convex set.

Definition . Dot x called boundary point set if the ε -neighborhood contains points belonging to the set X and points not belonging to the set X.

Definition . Dot x called internal point set, if it is found that the ε-neighborhood lies entirely inside the set X.

Comment . The boundary point may not belong to the set X. For example, for a set Definition. A bunch of X called limited, if its diameter is a finite number.

Definition . Cone is called a set such that it follows that .

Comment . From the definition it follows that the cone contains the zero point X= 0. The cone is an unbounded set (except for the degenerate case when the cone contains only one point X= 0). A cone can be either a closed or an open set.

Definition . Compact is called a closed bounded set.

Comment . Closed bounded sets are of particular interest in connection with Weierstrass's theorem, which states that continuous function on closed limited set(compact) reaches its highest and lowest values.

When researching economic phenomena mathematical methods Such a property of many sets and functions as convexity turns out to be very significant. The behavior of many economic entities is due to the fact that certain dependencies, describing these objects, are convex.

The convexity of functions and sets is often associated with the existence or uniqueness of solutions to economic problems: many computational algorithms are based on this same property.

The validity of many statements relating to convex sets and functions is completely clear, they are almost obvious. At the same time, their proof is often very difficult. Therefore, some basic facts related to convexity will be stated here, without proof, in the hope of their intuitive persuasiveness.

Convex sets on the plane.

Any geometric figure on the plane can be considered as a set of points belonging to this figure. Some sets (for example, a circle, a rectangle, a strip between parallel lines) contain both internal and boundary points; others (for example, a segment, a circle) consist only of boundary points.

A set of points on a plane is called convex if it has the following property: a segment connecting any two points of this set is entirely contained in this set.

Examples of convex sets are: triangle, segment, half-plane (part of a plane lying on one side of a line), the entire plane.

A set consisting of a single point and an empty set not containing a single point are, by convention, also considered convex. In any case, in these sets it is impossible to draw a segment connecting some points of these sets and not entirely belonging to these sets - it is generally impossible to select two points in them. Therefore, their inclusion in the number of convex sets will not lead to a contradiction with the definition, and this is sufficient for mathematical reasoning.

Intersection, i.e. a common part two convex sets, always convex: taking any two points of intersection (and they are common, i.e., they belong to each of the intersecting sets) and connecting them with a segment, we can easily verify that all points of the segment are common to both sets, so how each of them bulged. The intersection of any number of convex sets will also be convex.

An important property of convex sets is their separability: if two convex sets have no common internal points, then the plane can be cut along a straight line in such a way that one of the sets will lie entirely in one half-plane, and the other in the other (points of both sets can be located on the cut line). The straight line separating them in some cases turns out to be the only possible one, in others it is not.

The boundary point of any convex set can itself be considered as a convex set that does not have common internal points with the original set; therefore, it can be separated from it by some straight line. The straight line separating its boundary point from a convex set is called the supporting straight line of this set at a given point. The reference lines may be the only ones at some points of the contour, but not the only ones at others.

Let us introduce the system on the plane Cartesian coordinates x, y. Now we have the opportunity to consider various figures as sets of points whose coordinates satisfy certain equations or inequalities (if the coordinates of a point satisfy any condition, for brevity we will say that the point itself satisfies this condition).