Some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles.

Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids.

The collection of convex subsets of a vector space has the following properties:

# The union of a non-decreasing sequence of convex subsets is a convex set.

and A, B convex, non-empty subsets of V.

In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.

When X is a locally convex topological vector space, the continuous dual can be equipped with the strong topology, the topology of uniform convergence on bounded subsets of X.

However, no set of 4 points can be shattered: by Radon's theorem, any four points can be partitioned into two subsets with intersecting convex hulls, so it is not possible to separate one of these two subsets from the other.

* Radon's theorem, that d + 2 points in d dimensions may always be partitioned into two subsets with intersecting convex hulls

Then is defined to be the convex hull ( in ) of the barycentres of all subsets of the vertices of that contain.

There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations.

However, the situation is different for uniformly continuous functions defined on compact or convex subsets of normed spaces.

Then I and J form the required partition of the points into two subsets with intersecting convex hulls.

points in Euclidean d-space, there is a partition into r subsets whose convex hulls intersect in at least one common point.

In this more general context, the convex hull of a set S is the intersection of the family members that contain S, and the Radon number of a space is the smallest r such that any r points have two subsets whose convex hulls intersect.

With this definition, every set of ω + 1 vertices in the graph can be partitioned into two subsets whose convex hulls intersect, and ω + 1 is the minimum number for which this is possible, where ω is the clique number of the given graph.

is a finite collection of convex subsets of, where.

For infinite collections one has to assume compactness: If is a collection of compact convex subsets of and every subcollection of cardinality at most has nonempty intersection, then the whole collection has nonempty intersection.

( Without local convexity this is false, as the spaces for < math > 0 < p < 1 </ math > have no nontrivial open convex subsets.

Given a dual pair with a locally convex space and its continuous dual then is a dual topology on if and only if it is a topology of uniform convergence on a family of absolutely convex and weakly compact subsets of

In Euclidean space R < sup > n </ sup >, or any convex subset of R < sup > n </ sup >, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element.

* For a differentiable Lipschitz map ƒ: U → R < sup > m </ sup > the inequality holds for the best Lipschitz constant of f, and it turns out to be an equality if the domain U is convex.

In mathematics, Minkowski's theorem is the statement that any convex set in R < sup > n </ sup > which is symmetric with respect to the origin and with volume greater than 2 < sup > n </ sup > d ( L ) contains a non-zero lattice point.

Suppose that L is a lattice of determinant d ( L ) in the n-dimensional real vector space R < sup > n </ sup > and S is a convex subset of R < sup > n </ sup > that is symmetric with respect to the origin, meaning that if x is in S then − x is also in S.

Suppose that Γ is a lattice in n-dimensional Euclidean space R < sup > n </ sup > and K is a convex centrally symmetric body.

For any convex function f on R < sup > n </ sup >, one has

Let f < sub > 1 </ sub >, …, f < sub > m </ sub > be proper convex functions on R < sup > n </ sup >.

In the two-dimensional case the algorithm is also known as Jarvis's march, after R. A. Jarvis, who published it in 1973 ; it has O ( nh ) time complexity, where n is the number of points and h is the number of points on the convex hull.

An illustration of Carathéodory's theorem ( convex hull ) for a square in R < sup > 2 </ sup >.

A linear program may be specified by a system of real variables ( the coordinates for a point in Euclidean space R < sup > n </ sup >), a system of linear constraints ( specifying that the point lie in a halfspace ; the intersection of these halfspaces is a convex polytope, the feasible region of the program ), and a linear function ( what to optimize ).

In integral geometry ( otherwise called geometric probability theory ), Hadwiger's theorem characterises the valuations on convex bodies in R < sup > n </ sup >.

Let K < sup > n </ sup > be the collection of all convex bodies in R < sup > n </ sup >.

In convex geometry Carathéodory's theorem states that if a point x of R < sup > d </ sup > lies in the convex hull of a set P, there is a subset P ′ of P consisting of d + 1 or fewer points such that x lies in the convex hull of P ′.

The sample covariance matrix ( SCM ) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in R < sup > p × p </ sup >; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator.

Let f < sub > 1 </ sub >, …, f < sub > m </ sub > be proper convex functions on R < sup > n </ sup >.

In geometry, Radon's theorem on convex sets, named after Johann Radon, states that any set of d + 2 points in R < sup > d </ sup > can be partitioned into two ( disjoint ) sets whose convex hulls intersect.

Suppose Ω is an open, connected and convex subset of the Euclidean space R < sup > 2 </ sup > with smooth boundary ∂ Ω and suppose that D is the unit disk.

for < math > 0 < r < R ,</ math > then this function is strictly increasing and logarithmically convex.

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

A convex set.

A convex function | function is convex if and only if its Epigraph ( mathematics ) | epigraph, the region ( in green ) above its graph of a function | graph ( in blue ), is a convex set.

A set C in S is said to be convex if, for all x and y in C and all t in the interval, the point

is in C. In other words, every point on the line segment connecting x and y is in C. This implies that a convex set in a real or complex topological vector space is path-connected, thus connected.

A set C is called absolutely convex if it is convex and balanced.

If is a convex set, for any in, and any nonnegative numbers such that, then the vector

# The empty set and the whole vector-space are convex.

Every subset A of the vector space is contained within a smallest convex set ( called the convex hull of A ), namely the intersection of all convex sets containing A.

For a set of scattered points in the plane, the diameter of the points is the same as the diameter of their convex hull.

The problem of finding the Delaunay triangulation of a set of points in d-dimensional Euclidean space can be converted to the problem of finding the convex hull of a set of points in ( d + 1 )- dimensional space, by giving each point p an extra coordinate equal to | p |< sup > 2 </ sup >, taking the bottom side of the convex hull, and mapping back to d-dimensional space by deleting the last coordinate.

* The each triangle of the Delaunay triangulation of a set of points in d-dimensional spaces corresponds to a facet of convex hull of the projection of the points onto a ( d + 1 )- dimensional paraboloid, and vice versa.

After the invention of the telescope Kepler set out the theoretical basis on how they worked and described an improved version, known as the Keplerian telescope, using two convex lenses to produce higher magnification.

A simplex may be defined as the smallest convex set containing the given vertices.