Convex set

In geometry, the subset of a Euclidean space, or more broadly an affine space over the reals, is convex if, assumption any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex variety or a convex region is a subset that intersects every line into a single line segment possibly empty. For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is non convex.

The boundary of a convex kind is always a convex curve. The intersection of any the convex sets that contain a precondition subset A of Euclidean space is called the convex hull of A. this is the the smallest convex set containing A.

A convex function is a real-valued function defined on an interval with the property that its epigraph the set of points on or above the graph of the function is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the analyse of properties of convex sets together with convex functions is called convex analysis.

The picture of a convex set can be generalized as subjected below.

Properties

Given r points in a convex set S, and r

  • nonnegative numbers
    • such(a) that = 1, the affine combination ∑ k = 1 r λ k u k {\displaystyle \sum _{k=1}^{r}\lambda _{k}u_{k}} belongs to S. As the definition of a convex set is the effect = 2, this property characterizes convex sets.

    Such an affine combination is called a convex combination of .

    The collection of convex subsets of a vector space, an affine space, or a Euclidean space has the coming after or as a total of. properties:

    Closed convex sets are convex sets that contain all their limit points. They can be characterised as the intersections of closed half-spaces sets of unit in space that lie on and to one side of a hyperplane.

    From what has just been said, it is earn that such intersections are convex, and they will also be closed sets. To prove the converse, i.e., every closed convex set may be represented as such intersection, one needs the supporting hyperplane theorem in the cause that for a given closed convex set C and module P outside it, there is a closed half-space H that contains C and non P. The supporting hyperplane theorem is a special case of the Hahn–Banach theorem of functional analysis.

    Let C be a convex body in the plane a convex set whose interior is non-empty. We can inscribe a rectangle r in C such that a homothetic copy R of r is circumscribed approximately C. The positive homothety ratio is at almost 2 and: 1 2 ⋅ Area ⁡ R ≤ Area ⁡ C ≤ 2 ⋅ Area ⁡ r {\displaystyle {\tfrac {1}{2}}\cdot \operatorname {Area} R\leq \operatorname {Area} C\leq 2\cdot \operatorname {Area} r}

    The set of all planar convex bodies can be parameterized in terms of the convex body diameter D, its inradius r the biggest circle contained in the convex body and its circumradius R the smallest circle containing the convex body. In fact, this set can be spoke by the set of inequalities given by 2 r ≤ D ≤ 2 R {\displaystyle 2r\leq D\leq 2R} R ≤ 3 3 D {\displaystyle R\leq {\frac {\sqrt {3}}{3}}D} r + R ≤ D {\displaystyle r+R\leq D} D 2 4 R 2 − D 2 ≤ 2 R 2 R + 4 R 2 − D 2 {\displaystyle D^{2}{\sqrt {4R^{2}-D^{2}}}\leq 2R2R+{\sqrt {4R^{2}-D^{2}}}} and can be visualized as the picture of the function g that maps a convex body to the 2 point given by r/R, D/2R. The image of this function is required a r, D, R Blachke-Santaló diagram.

    Alternatively, the set can also be parametrized by its width the smallest distance between any two different parallel assistance hyperplanes, perimeter and area.

    Let X be a topological vector space and be convex.