Convex set

In mathematics, ie, a geometric figure, or more generally a subset of a Euclidean space is convex if for any two arbitrary points, which belong to the set, and is always the link entirely in the crowd. This guarantees that the amount has a ( concave ) indentation at any point.

History and application

The theory of convex sets founded Hermann Minkowski in his work, the geometry of numbers, Leipzig 1910. Application find convex sets eg in convex optimization, or computer animation, where convex polytopes are easier to handle in different ways than Nonconvex.

Definition of vector spaces

A subset of a real or complex vector space is called convex if for all and for all with always:

This definition is based on the parametric representation of the link between and:

In fact, the above definition also includes objects with planar edges such as squares with one that you would not necessarily colloquially called convex.

Examples

• Each vector space containing is convex, as the half-planes and half-spaces.
• Example subsets of figurative Euclidean space: The empty set and every singleton are convex.
• Finite sets are convex if and only if they contain at most one element.
• Routes and lines are convex sets.
• Each triangular face and all regular polygonal surfaces are convex.
• Circular disks and spheres are convex.
• Among the four corners are eg the convex parallelograms, trapezoids and kites while there are squares that are nichtkonvex how the crossed trapezoid or the arrow quadrangle.
• Cube, platonic solids, and spars are convex.
• The partial quantity is above or below the graph of a convex or concave function, is convex.
• A torus ( donut ) is not convex.
• The boundary of a convex set is nichtkonvex in general.

Properties

• Each non-empty convex subset of a real or complex vector space is connected, contractible to a point and thus can not have any holes.

• The average of any (even infinite) many convex sets is convex. Thus, the convex subsets of a vector space form a containment system. The union of convex sets, however, is not convex in general.

• The convex hull of a set is the smallest convex superset. Is the average of all convex quantities in which it is contained.

• In locally convex spaces is a compact convex set of financial statements of convex combinations of its extreme points ( Krein - Milman ). Here, a extreme point is a point which is not located between two points.

• Every convex set is star-shaped, such that each point can be selected as the star center.

Generalizations

General sufficient for meaningful definition of convexity already considerably weaker assumptions on the geometry that applies to, it takes from Hilbert's system of axioms of Euclidean geometry only the axioms of logic and of the arrangement. The convexity is particularly dependent on the definition of a straight link. Thus, the half-plane which is defined by is convex in the Euclidean plane, but nichtkonvex in the Moulton plane: for example, runs the "straight line" between and above the (not in the quantity contained ) point. See also collinear.

Depending on the mathematical context different generalizations are used that are not even partially coherent.

Konvexitätsraum

Following axioms generalized the basic properties of convex sets on a level which is similar to the topology.

A lot together with a set of subsets is called Konvexitätsraum when applies to the following:

• The empty set and itself are in
• The intersection of any number of quantities is again in
• If a subset is totally ordered with respect to inclusion, so is the union of all sets in from.

Then the amounts of the convex sets of are called.

Metric convex space

A metric space is called metrically convex if for every two points always exists an ( intermediate ) point so that equality holds in the triangle inequality:

Here, however, is no longer true that the intersection of convex sets metric would be metrically convex again. Thus, the circle with the metric of the arc length metric is convex, two closed half- circles that are disjoint except for their two end points, also metrically convex ( part ) quantities, their two-element section are not.

Geodetically convex manifolds

Semi- Riemannian manifolds have an inherent metric that determines the geodesics of the manifold. If each pair of points can be connected in an environment with a single geodesic of the manifold, which is completely in this environment, this environment is called simply convex.

A submanifold of a Riemannian manifold is called geodesically convex if any two points can be represented by a curve in Connect, which is a length -minimizing geodesic in globally.

Examples and differences

• The rational number with the usual distance metric form a convex subset of that is not convex.
• The same applies to what is not geodesically convex Riemannian manifold as.
• A convex subset of Euclidean space is always also metrically convex, with respect to the induced by the standard metric. For closed subsets of the converse also holds.

Curvature of curves

In two dimensions, the curvature of a continuously differentiable curve in a point in relation to the observer can be studied:

• If the neighboring points of the same tangential half-plane as the viewer, it is concave there for him.
• There is an environment, such that all of the points thereof lie in the other half of the tangential plane, the curve is convexly curved in the viewer.

Corners are called convex if all interior angles exceed 180 °.

Analog can be studied in higher dimensions the curvature of hyperplanes, to the object must be orientable, however.