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16 relations: Carathéodory's theorem (convex hull), Centerpoint (geometry), Choquet theory, Convex hull, Convex polygon, Convex set, Eduard Helly, Family of sets, Geometric separator, Helly family, Krein–Milman theorem, List of convexity topics, List of theorems, Nina Amenta, Oriented matroid, Radon's theorem.
Carathéodory's theorem (convex hull)
In convex geometry, Carathéodory's theorem states that if a point x of Rd lies in the convex hull of a set P, then x can be written as the convex combination of at most d + 1 points in P. Namely, there is a subset ′ of P consisting of d + 1 or fewer points such that x lies in the convex hull of ′.
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Centerpoint (geometry)
In statistics and computational geometry, the notion of centerpoint is a generalization of the median to data in higher-dimensional Euclidean space.
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Choquet theory
In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral taken over the set E of extreme points.
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Convex hull
In mathematics, the convex hull or convex envelope or convex closure of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X., p. 3.
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Convex polygon
A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon.
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Convex set
In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.
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Eduard Helly
Eduard Helly (June 1, 1884, Vienna – 1943, Chicago) was a mathematician after whom Helly's theorem, Helly families, Helly's selection theorem, Helly metric, and the Helly–Bray theorem were named.
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Family of sets
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.
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Geometric separator
A geometric separator is a line (or other shape) that partitions a collection of geometric shapes into two subsets, such that proportion of shapes in each subset is bounded, and the number of shapes that do not belong to any subset (i.e. the shapes intersected by the separator itself) is small.
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Helly family
In combinatorics, a Helly family of order k is a family of sets such that any minimal subfamily with an empty intersection has k or fewer sets in it.
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Krein–Milman theorem
In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about convex sets in topological vector spaces.
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List of convexity topics
This is a list of convexity topics, by Wikipedia page.
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List of theorems
This is a list of theorems, by Wikipedia page.
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Nina Amenta
Annamaria Beatrice (Nina) Amenta is an American computer scientist who works as the Tim Bucher Family Professor of Computer Science and the chair of the Computer Science Department at the University of California, Davis.
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Oriented matroid
An oriented matroid is a mathematical structure that abstracts the properties of directed graphs and of arrangements of vectors in a vector space over an ordered field (particularly for partially ordered vector spaces).
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Radon's theorem
In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that any set of d + 2 points in Rd can be partitioned into two disjoint sets whose convex hulls intersect.
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