Convex set

Convex set.

Convex set.

A convex set in an affine or vector space is a set in which all points of a segment formed by any two points of a given set also belong to this set.

Content

Definitions

Let be $A$ ${\ displaystyle A}$ - affine or vector space over the field of real numbers $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ .

Lots of $K\subset A$ ${\ displaystyle K \ subset A}$ called convex if, together with any two points $x,\;y\in K$ ${\ displaystyle x, \; y \ in K}$ many $K$ ${\ displaystyle K}$ all points of the segment belong $x y$ ${\ displaystyle xy}$ connecting in space $A$ ${\ displaystyle A}$ points $x$ ${\ displaystyle x}$ and $y$ ${\ displaystyle y}$ . This segment can be represented as

\bigcup \limits _{t\in [0;\;1]}\{x+t\cdot {\overrightarrow {xy}}\}.

{\ displaystyle \ bigcup \ limits _ {t \ in [0; \; 1]} \ {x + t \ cdot {\ overrightarrow {xy}} \}.}

Related Definitions

Lots of $K$ ${\ displaystyle K}$ vector space $V$ ${\ displaystyle V}$ is called absolutely convex if it is convex and balanced .

Examples

Convex subsets of sets $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ (a set of real numbers) are intervals of $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ .
Examples of convex subsets in two - dimensional Euclidean space ( $\mathbb {R} ^{2}$ ${\ displaystyle \ mathbb {R} ^ {2}}$ ) are regular polygons .
Examples of convex subsets in three-dimensional Euclidean space ( $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ ) are Archimedean bodies 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 linearly 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 be $K$ ${\ displaystyle K}$ Is a convex set in linear space. Then for any elements $u_{1},\;u_{2},\;\ldots ,\;u_{r}$ ${\ displaystyle u_ {1}, \; u_ {2}, \; \ ldots, \; u_ {r}}$ owned by $K$ ${\ displaystyle K}$ and for all non-negative $\lambda _{1},\;\lambda _{2},\;\ldots ,\;\lambda _{r}$ ${\ displaystyle \ lambda _ {1}, \; \ lambda _ {2}, \; \ ldots, \; \ lambda _ {r}}$ such that $\lambda _{1}+\lambda _{2}+\ldots +\lambda _{r}=1$ ${\ displaystyle \ lambda _ {1} + \ lambda _ {2} + \ ldots + \ lambda _ {r} = 1}$ , vector
$w=\sum _{k=1}^{r}\lambda _{k}u_{k}$ ${\ displaystyle w = \ sum _ {k = 1} ^ {r} \ lambda _ {k} u_ {k}}$

belongs

K

{\ displaystyle K}

.

Vector $w$ ${\ displaystyle w}$ called a convex combination of elements $u_{1},\;u_{2},\;\ldots ,\;u_{r}$ ${\ displaystyle u_ {1}, \; u_ {2}, \; \ ldots, \; u_ {r}}$ .

The intersection of any number of convex sets is a convex set, so convex subsets form a complete grid ^{[ unknown term ]} . It follows that for any subset $A$ ${\ displaystyle A}$ linear space there exists the smallest convex set containing it. This set is the intersection of all convex sets containing $A$ ${\ displaystyle A}$ and is called the convex hull of the set $A$ ${\ displaystyle A}$ .
Closed convex sets can be defined as the intersections of closed half - spaces (sets of points in space that lie on only one part of the hyperplane ). From the foregoing, it becomes clear that such intersections are convex and closed sets. To prove the opposite, that is, that each convex set can be represented as an intersection, we can use the support hyperplane theorem in the form in which for a given closed convex set $C$ ${\ displaystyle C}$ and points $P$ ${\ displaystyle P}$ that does not belong to him, there is a closed half-space $H$ ${\ displaystyle H}$ containing $C$ ${\ displaystyle C}$ and not containing $P$ ${\ displaystyle P}$ . The support hyperplane theorem is a special case of the Hahn - Banach theorem from functional analysis .
Helly's theorem : Suppose in a finite family of convex subsets $\mathbb {R} ^{d}$ ${\ displaystyle \ mathbb {R} ^ {d}}$ intersection of any $d+1$ ${\ displaystyle d + 1}$ of which is nonempty. Then the intersection of all subsets of this family is nonempty.
Any convex set of unit area in $\mathbb {R} ^{2}$ ${\ displaystyle \ mathbb {R} ^ {2}}$ can be entirely enclosed in some triangle of area 2 ^[1] .

Variations and generalizations

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

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. Combinatorics of convex polyhedra . - M .: MCCNMO , 2002 .-- 16 p. - ISBN 5-94057-024-0 . .

Links

↑ Weisstein, Eric W. Triangle Circumscribing on the Wolfram MathWorld website.

[1] Weisstein, Eric W. Triangle Circumscribing on the Wolfram MathWorld website.