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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 ...
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.
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.
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.
The tuple (K;n,R,r) is called a well-bounded convex set. An interior point a 0, which is a point such that a ball of radius r around a 0 is contained in K. The tuple (K;n,R,r,a 0) is called a centered convex set. The following can be done in oracle-polynomial time: [1]: Sec.4 An oracle for WSEP, with a circumscribed radius R, can solve WVIOL.
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 ...
This generalizes the idea of convexity in Euclidean geometry, where given two points , in a convex set , all of the points + are contained in that set. There is a vector field X U p {\displaystyle {\mathcal {X}}_{U_{p}}} in a neighborhood U p {\displaystyle U_{p}} of p {\displaystyle p} transporting p {\displaystyle p} to each point p ′ ∈ ...
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 ...