Search results
Results From The WOW.Com Content Network
Equivalently, a convex set or a convex region is a set that intersects every line in a line segment, single point, or the empty set. [1] [2] For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set in the plane is always a convex curve.
We prove the finite version, using Radon's theorem as in the proof by Radon (1921).The infinite version then follows by the finite intersection property characterization of compactness: a collection of closed subsets of a compact space has a non-empty intersection if and only if every finite subcollection has a non-empty intersection (once you fix a single set, the intersection of all others ...
In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set.
Radon's theorem forms a key step of a standard proof of Helly's theorem on intersections of convex sets; [7] this proof was the motivation for Radon's original discovery of Radon's theorem. Radon's theorem can also be used to calculate the VC dimension of d-dimensional points with respect to linear separations.
The set T is also called a rainbow simplex, since it is a d-dimensional simplex in which each corner has a different color. [12] This theorem has a variant in which the convex hull is replaced by the conical hull. [10]: Thm.2.2 Let X 1, ..., X d be sets in R d and let x be a point contained in the intersection of the conical hulls of all these ...
A set in ℝ 2 satisfying the hypotheses of Minkowski's theorem. In mathematics , Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to the origin and which has volume greater than 2 n {\displaystyle 2^{n}} contains a non-zero integer point (meaning a point in Z n ...
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space.There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap.
A convex set can have more than one supporting hyperplane at a given point on its boundary. This theorem states that if S {\displaystyle S} is a convex set in the topological vector space X = R n , {\displaystyle X=\mathbb {R} ^{n},} and x 0 {\displaystyle x_{0}} is a point on the boundary of S , {\displaystyle S,} then there exists a ...