convex set

In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. Connectedness is one of the principal topological properties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, which is a space where any two points can be joined by a path. It is usually easy to think about what is not connected. A simple example would be a space consisting of two rectangles, each of whic ...

Including:

• Connected space - Formal definition
• Connected space - Examples
• Connected space - Path connectedness
• Connected space - Local connectedness
• Connected space - Theorems

Read more here: » Connected space: Encyclopedia - Connected space

In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. (Thus, an object is, among other things, a set.) An object is closed if it is equal to its closure. Typical structural properties of all closure operations are: The closure is increasing or extensive: the closure of an object contains the object. The closure is idempotent: the closure of the closure equals t ...

Including:

• Closure mathematics - Examples

Read more here: » Closure mathematics: Encyclopedia - Closure mathematics

A subset C of a vector space V is a convex cone iff αx + βy belongs to C, for any positive scalars α, β of V, and any x, y in C. The defining condition can be written more succintly as "αC + βC = C for any positive scalars α, β of V. The concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over the rational, algeb ...

See also:

Convex cone, Convex cone - Definition, Convex cone - Convex cones are linear cones, Convex cone - Alternative definitions, Convex cone - Blunt and pointed cones, Convex cone - Half-spaces, Convex cone - Salient convex cones and perfect half-spaces, Convex cone - Cross-sections and projections of a convex set, Convex cone - Flat section, Convex cone - Spherical section, Convex cone - Partial order defined by a convex cone, Convex cone - Proper convex cone

Read more here: » Convex cone: Encyclopedia II - Convex cone - Definition

In computational geometry, numerous algorithms are proposed for computing the convex hull of a finite set of points, with various computational complexity. Computing the convex hull means that a non-ambiguous and efficient representation of the required convex shape is constructed. The complexity of the corresponding algorithms is usually estimated in terms of n, the number of input points, and h, the number of points on the convex hull. ...

See also:

Convex hull, Convex hull - Alternative definitions, Convex hull - Intuitive picture, Convex hull - Computation of convex hulls, Convex hull - Planar case, Convex hull - Higher dimensions, Convex hull - Relations to other geometric structures, Convex hull - Applications

Read more here: » Convex hull: Encyclopedia II - Convex hull - Computation of convex hulls

A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors specifically exclude the empty set with its unique topology as a connected space, but this encyclopedia does not follow that practice. For a topological space X the following conditions are equivalent: X is connected.

A convex function f defined on some convex open interval C is continuous on C and differentiable at all but at most countably many points. If C is closed, then f may fail to be continuous at the endpoints of C. A continuous function on an interval C is convex if and only if for any x and y in C. A differentiable function of one variable is convex on an interval if and only i ...

See also:

Convex function, Convex function - Properties of convex functions, Convex function - Examples

Read more here: » Convex function: Encyclopedia II - Convex function - Properties of convex functions

If a simple polygon is not convex, it is called concave. At least one internal angle of a concave polygon is larger than 180 degrees. A concave polygon is often called re-entrant polygon (but in some cases the latter term has a different meaning). ...

See also:

Convex polygon, Convex polygon - Concave polygons

Read more here: » Convex polygon: Encyclopedia II - Convex polygon - Concave polygons

For planar objects, i.e., lying in the plane, an easy way to visualize the convex hull is to imagine a rubber band tightly stretched to encompass the given objects; when released, it will assume the shape of the required convex hull. An attempt may be made to generalise this picture by imagining some objects contained in a sort of idealised unpressurised rubber balloon under tension . However, the minimum energy surface in this case may not be the convex hull - parts of the resulting surface may have negative curvature, like a ...

See also:

Convex hull, Convex hull - Alternative definitions, Convex hull - Intuitive picture, Convex hull - Computation of convex hulls, Convex hull - Planar case, Convex hull - Higher dimensions, Convex hull - Relations to other geometric structures, Convex hull - Applications

Read more here: » Convex hull: Encyclopedia II - Convex hull - Intuitive picture

The space X is said to be path-connected if for any two points x and y in X there exists a continuous function f from the unit interval [0,1] to X with f(0) = x and f(1) = y. (This function is called a path from x to y.) Every path-connected space is connected. Example of connected spaces that are not path-connected include the extended long ...

See also:

Connected space, Connected space - Formal definition, Connected space - Examples, Connected space - Path connectedness, Connected space - Local connectedness, Connected space - Theorems

Read more here: » Connected space: Encyclopedia II - Connected space - Path connectedness

In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X. (Note that X may be the union of any set of objects made of points). To show this exists, it is necessary to see that every X is contained in at least one convex set (the whole space V, for example), and any intersection of convex sets containing X is also a convex set containing X. It is then clear that the convex hull is the intersection of all convex ...

See also:

Convex hull, Convex hull - Alternative definitions, Convex hull - Intuitive picture, Convex hull - Computation of convex hulls, Convex hull - Planar case, Convex hull - Higher dimensions, Convex hull - Relations to other geometric structures, Convex hull - Applications

Read more here: » Convex hull: Encyclopedia II - Convex hull - Alternative definitions

A convex cone is said to be flat if it contains some nonzero vector x and its opposite -x; and salient otherwise. A blunt convex cone is necessarily salient, but the converse is not necessarily true. A convex cone C is salient if and only if C(-C) is {0}; that is, iff C does not contain any non-trivial linear subspace of V. A perfect half-space of V is defined recursively as follows: if V is zero-dimensional, then it is the set {0}, else it is any open half-space H of V ...

See also:

Convex cone, Convex cone - Definition, Convex cone - Convex cones are linear cones, Convex cone - Alternative definitions, Convex cone - Blunt and pointed cones, Convex cone - Half-spaces, Convex cone - Salient convex cones and perfect half-spaces, Convex cone - Cross-sections and projections of a convex set, Convex cone - Flat section, Convex cone - Spherical section, Convex cone - Partial order defined by a convex cone, Convex cone - Proper convex cone

Read more here: » Convex cone: Encyclopedia II - Convex cone - Salient convex cones and perfect half-spaces

A topological space is said to be locally connected if it has a base of connected sets. It can be shown that a space X is locally connected if and only if every component of every open set of X is open. The topologist's sine curve shown above is an example of a connected space that is not locally connected. Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connec ...

See also:

Connected space, Connected space - Formal definition, Connected space - Examples, Connected space - Path connectedness, Connected space - Local connectedness, Connected space - Theorems

Read more here: » Connected space: Encyclopedia II - Connected space - Local connectedness

An hyperplane of V is a maximal proper linear subspace of V. An open (resp. closed) half-space of V is any subset H of V defined by the condition L(x) > 0 (resp. L(x)0), where L is any linear function from V to its scalar field. The hyperplane defined by L(v) = 0 is the bounding hyperplane of H. Half-spaces (open or closed) are convex cones. Moreover, any convex cone C that is not the whole spa ...

See also:

Convex cone, Convex cone - Definition, Convex cone - Convex cones are linear cones, Convex cone - Alternative definitions, Convex cone - Blunt and pointed cones, Convex cone - Half-spaces, Convex cone - Salient convex cones and perfect half-spaces, Convex cone - Cross-sections and projections of a convex set, Convex cone - Flat section, Convex cone - Spherical section, Convex cone - Partial order defined by a convex cone, Convex cone - Proper convex cone

Read more here: » Convex cone: Encyclopedia II - Convex cone - Half-spaces

Convex cone - Flat section. An affine hyperplane of V is any subset of V of the form v + H, where v is a vector of V and H is a (linear) hiperplane. The following result follows from the property of containment by half-spaces. Let Q be an open half-space of V, and A = H + v where H is the bounding hyperplane of Q and v is any vector in Q. Let C be a linear cone contained in Q ...

See also:

Convex cone, Convex cone - Definition, Convex cone - Convex cones are linear cones, Convex cone - Alternative definitions, Convex cone - Blunt and pointed cones, Convex cone - Half-spaces, Convex cone - Salient convex cones and perfect half-spaces, Convex cone - Cross-sections and projections of a convex set, Convex cone - Flat section, Convex cone - Spherical section, Convex cone - Partial order defined by a convex cone, Convex cone - Proper convex cone

Read more here: » Convex cone: Encyclopedia II - Convex cone - Cross-sections and projections of a convex set