On the existence of hyperplanes separating disjoint convex sets
Hyperplane separation theorem
Illustration of the hyperplane separation theorem.
Type
Theorem
Field
Convex geometry
Topological vector spaces
Collision detection
Conjectured by
Hermann Minkowski
Open problem
No
Generalizations
Hahn–Banach separation theorem
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. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint.
The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces.
A related result is the supporting hyperplane theorem.
In the context of support-vector machines, the optimally separating hyperplane or maximum-margin hyperplane is a hyperplane which separates two convex hulls of points and is equidistant from the two.[1][2][3]
^Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome (2008). The Elements of Statistical Learning : Data Mining, Inference, and Prediction(PDF) (Second ed.). New York: Springer. pp. 129–135.
^Witten, Ian H.; Frank, Eibe; Hall, Mark A.; Pal, Christopher J. (2016). Data Mining: Practical Machine Learning Tools and Techniques (Fourth ed.). Morgan Kaufmann. pp. 253–254. ISBN 9780128043578.
^Deisenroth, Marc Peter; Faisal, A. Aldo; Ong, Cheng Soon (2020). Mathematics for Machine Learning. Cambridge University Press. pp. 337–338. ISBN 978-1-108-45514-5.
and 22 Related for: Hyperplane separation theorem information
In geometry, the hyperplaneseparationtheorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar...
separated by a hyperplane, a result called the hyperplaneseparationtheorem. In machine learning, hyperplanes are a key tool to create support vector machines...
subset containing one but disjoint from the other. Hyperplaneseparationtheorem - either of two theorems about disjoint convex sets in n-dimensional Euclidean...
half-space is the half-space that includes the points within the hyperplane. This theorem states that if S {\displaystyle S} is a convex set in the topological...
corners of the polygon to recover the entire polygon shape. Hyperplaneseparationtheorem: Any two convex polygons with no points in common have a separator...
the decision boundary and data points. Discriminant function Hyperplaneseparationtheorem Corso, Jason J. (Spring 2013). "Quiz 1 of 14 - Solutions" (PDF)...
pair of the element and the index of the set that contains it. Hyperplaneseparationtheorem for disjoint convex sets Mutually exclusive events Relatively...
total weight can be separated. See also: Geometric separator Hyperplaneseparationtheorem Some recently-studied variants of the problem include: Guillotine-cutting...
the smallest convex cone containing C (a consequence of the hyperplaneseparationtheorem) A cone C in a vector space X is said to be self-dual if X can...
versions of the hyperplaneseparationtheorem are also known (in German) as Trennungssatz von Eidelheit (Eidelheit separationtheorem). A theorem on the solubility...
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety...
general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective...
find a point in K. Strong separation problem (SSEP): given a vector y in Rn, decide whether y in K, and if not, find a hyperplane that separates y from K...
(n − 1)-dimensional hyperplane. Guillotine separation: the problem of separating convex objects in the plane using guillotine cuts. Other Separationtheorems. Simultaneous...
arrangements of hyperplanes). Many results—Carathéodory's theorem, Helly's theorem, Radon's theorem, the Hahn–Banach theorem, the Krein–Milman theorem, the lemma...
{\displaystyle S,} then there exists a supporting hyperplane containing x . {\displaystyle x.} The hyperplane in the theorem may not be unique, as noticed in the second...
positive examples cannot be separated from the negative examples by a hyperplane, then the algorithm would not converge since there is no solution. Hence...
projective spaces Adequate equivalence relation Hilbert scheme Lefschetz hyperplanetheorem Minimal model program Kollár & Moduli, Ch I. Shafarevich, Igor R....
called a norming functional for x . {\displaystyle x.} The Hahn–Banach separationtheorem states that two disjoint non-empty convex sets in a real Banach space...
Global Information