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.
Then () is a strict subset of that is not even convex; in particular, this example also shows that the balanced hull of a convex set is not necessarily convex. The set is equal to the closed and filled square in with vertices (,), (,), (,), and (,) (this is because the balanced set must contain both and = {(,), (,)}, where ...
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.
Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap .
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 ...
As a particular example, every convex combination of two points lies on the line segment between the points. [1] A set is convex if it contains all convex combinations of its points. The convex hull of a given set of points is identical to the set of all their convex combinations. [1]
According to the Krein–Milman theorem, every compact convex set in a Euclidean space (or more generally in a locally convex topological vector space) is the convex hull of its extreme points. [15] However, this may not be true for convex sets that are not compact; for instance, the whole Euclidean plane and the open unit ball are both convex ...
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 ...