Searches , social queries for CONVEX SET
Search references for CONVEX SET. Phrases containing CONVEX SET
See searches and references containing CONVEX SET!
Search references for CONVEX SET. Phrases containing CONVEX SET
See searches and references containing CONVEX SET!CONVEX SET
In geometry, set whose intersection with every line is a single line segment
geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube is a convex set, but anything
Convex_set
Smallest convex set containing a given set
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined
Convex_hull
Mathematical set with an ordering
convex sets of geometry, one uses order-convex instead of "convex". A convex sublattice of a lattice L is a sublattice of L that is also a convex set
Partially_ordered_set
Real function with secant line between points above the graph itself
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
Convex_function
Convex hull of a finite set of points in a Euclidean space
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n {\displaystyle n} -dimensional
Convex_polytope
problems in mathematical programming can be formulated as problems on convex sets or convex bodies. Six kinds of problems are particularly important: optimization
Algorithmic problems on convex sets
Algorithmic_problems_on_convex_sets
Polygon that is the boundary of a convex set
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is
Convex_polygon
Mathematical set closed under positive linear combinations
combinations with positive coefficients. It follows that convex cones are convex sets. The definition of a convex cone makes sense in a vector space over any ordered
Convex_cone
Convex and balanced set
disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set. A subset S {\displaystyle S} of a real
Absolutely_convex_set
Subfield of mathematical optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently
Convex_optimization
Mathematics of convex functions and sets
Convex analysis is the branch of mathematics that studies convex sets, convex functions, and their applications to optimization, functional analysis,
Convex_analysis
Property of point sets in Euclidean spaces
a set S {\displaystyle S} in the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is called a star domain (or star-convex set, star-shaped set or
Star_domain
Minimal superset that intersects each axis-parallel line in an interval
In geometry, a set K ⊂ Rd is defined to be orthogonally convex if, for every line L that is parallel to one of standard basis vectors, the intersection
Orthogonal_convex_hull
Type of metric space in mathematics
dimensions—are convex metric spaces. Given any two distinct points x {\displaystyle x} and y {\displaystyle y} in such a space, the set of all points z
Convex_metric_space
Type of plane curve
Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions. Important subclasses of convex curves
Convex_curve
Space with topology generated by convex sets
whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of
Locally convex topological vector space
Locally_convex_topological_vector_space
Branch of geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational
Convex_geometry
convex sets (POCS), sometimes known as the alternating projection method, is a method to find a point in the intersection of two closed convex sets.
Projections_onto_convex_sets
Type of mathematical functions
logarithmically convex. A Reinhardt domain D is called logarithmically convex if the image λ ( D ∗ ) {\displaystyle \lambda (D^{*})} of the set D ∗ = { z =
Function of several complex variables
Function_of_several_complex_variables
Linear combination of points
example, every convex combination of two points lies on the line segment between the points. A set is convex if it contains all convex combinations of
Convex_combination
Form of projection
used to solve non-differentiable convex optimization problems. Many interesting problems can be formulated as convex optimization problems of the form
Proximal_gradient_method
Point not between two other points
In mathematics, an extreme point of a convex set S {\displaystyle S} in a real or complex vector space or affine space is a point in S {\displaystyle S}
Extreme_point
An integrally convex set is the discrete geometry analogue of the concept of convex set in geometry. A subset X of the integer grid Z n {\displaystyle
Integrally_convex_set
Topics referred to by the same term
joins points Convex polygon, a polygon which encloses a convex set of points Convex polytope, a polytope with a convex set of points Convex metric space
Convex
Shape with seven sides
double lattice packing density of any convex set, and more generally for the optimal packing density of any convex set. Some 1000-kwacha coins from Zambia
Heptagon
All numbers between two given numbers
of the intervals. The concepts of convex sets and convex components are used in a proof that every totally ordered set endowed with the order topology is
Interval_(mathematics)
Sum of terms, each multiplied with a scalar
convex cones, and convex sets are generalizations of vector subspaces: a vector subspace is also an affine subspace, a convex cone, and a convex set,
Linear_combination
Shape that blocks all lines of sight
opaque set for the square, and for most other shapes this problem similarly remains unsolved. The shortest opaque set for any bounded convex set in the
Opaque_set
Distance from origin of tangent hyperplanes
In mathematics, the support function hA of a non-empty closed convex set A in R n {\displaystyle \mathbb {R} ^{n}} describes the (signed) distances of
Support_function
Mathematical function with convex lower level sets
a convex subset of a real vector space, such that for any real number y, the set of points on which the function value is at most y is a convex set. In
Quasiconvex_function
Points with no three in a line
spaces as well as from compact convex co-convex subsets of a convex set. An example of cap sets comes from the card game Set, a card game in which each card
Cap_set
Cone of outward normals to a convex set at a point
In convex analysis and optimization, the normal cone to a set at a point is a convex cone consisting of vectors that make a non-acute angle with every
Normal_cone_(convex_analysis)
Topics referred to by the same term
enclosing a strictly convex set of points Strictly convex set, a set whose interior contains the line between any two points Strictly convex space, a normed
Strictly_convex
Non-empty convex set in Euclidean space
mathematics, a convex body in n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is a compact convex set with non-empty
Convex_body
Set that can be "inflated" to reach any point
set – Convex and balanced set Balanced set – Construct in functional analysis Bornivorous set – Set that can absorb any bounded subset Bounded set (topological
Absorbing_set
Sums vector sets A and B by adding each vector in A to each vector in B
the Minkowski sum with a vector subtraction. If the two convex shapes intersect, the resulting set will contain the origin. A − B = { a − b | a ∈ A , b
Minkowski_addition
On when a space equals the closed convex hull of its extreme points
compact convex sets in locally convex topological vector spaces (TVSs). Krein–Milman theorem—A compact convex subset of a Hausdorff locally convex topological
Krein–Milman_theorem
Algorithms for solving convex optimization problems
convex function and G is a convex set. Without loss of generality, we can assume that the objective f is a linear function. Usually, the convex set G
Interior-point_method
Normed vector space for which the closed unit ball is strictly convex
strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space
Strictly_convex_space
Type of mathematical space
mathematics, a convex space (or barycentric algebra) is a space in which it is possible to take convex combinations of any finite set of points. A convex space
Convex_space
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
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
Minkowski's_theorem
Hyperplane in geometry
the hyperplane. This theorem states that if S {\displaystyle S} is a convex set in the topological vector space X = R n , {\displaystyle X=\mathbb {R}
Supporting_hyperplane
(1969) for a full treatment of this problem. When the set A {\displaystyle A} is a convex set, the lim-inf above is a true limit, and one can show that
Minkowski–Steiner_formula
computational geometry, a set of points in the Euclidean plane or a higher-dimensional Euclidean space is said to be in convex position or convex independent if
Convex_position
Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P
in convex geometry. It states that if a point x {\displaystyle x} lies in the convex hull C o n v ( P ) {\displaystyle \mathrm {Conv} (P)} of a set P ⊂
Carathéodory's theorem (convex hull)
Carathéodory's_theorem_(convex_hull)
Significant topic in economics
the intersection of two convex sets is a convex set. More generally, the intersection of a family of convex sets is a convex set. For every subset Q of
Convexity_in_economics
Violations of the convexity assumptions of elementary economics
convex preferences (that do not prefer extremes to in-between values) and convex budget sets and on producers with convex production sets; for convex
Non-convexity_(economics)
Problem in geometric probability
) Among continuous uniform distributions over bounded convex sets the probability of a convex quadrilateral is maximized by any circle or ellipse (probability
Sylvester's four point problem
Sylvester's_four_point_problem
Planar surface that forms part of the boundary of a solid object
According to this definition, the set of faces of a polytope includes the polytope itself and the empty set. For convex polytopes, this definition is equivalent
Face_(geometry)
geometry and computational geometry, the relative convex hull or geodesic convex hull is an analogue of the convex hull for the points inside a simple polygon
Relative_convex_hull
Concept in economics
preference relation ⪰ {\displaystyle \succeq } on the consumption set X is called convex if whenever x , y , z ∈ X {\displaystyle x,y,z\in X} where y ⪰ x
Convex_preferences
\sum _{i=1}^{\infty }r_{i}x_{i}} is called a convex series with elements of S {\displaystyle S} . If the set { x 1 , x 2 , … } {\displaystyle \left\{x_{1}
Convex_series
Shape containing unit line segments in all directions
360°. This question was first posed, for convex regions, by Sōichi Kakeya (1917). The minimum area for convex sets is achieved by an equilateral triangle
Kakeya_set
Haar null set Convex set Balanced set, Absolutely convex set Fractal set Recursive set Recursively enumerable set Arithmetical set Diophantine set Hyperarithmetical
List_of_types_of_sets
Optimization algorithm
the intersection of convex sets, and is a variant of the alternating projection method (also called the projections onto convex sets method). In its simplest
Dykstra's projection algorithm
Dykstra's_projection_algorithm
Convex polyhedron with regular faces
"cut-and-paste" manipulations of uniform solids. A convex polyhedron is the convex hull of a finite set of points in 3-dimensional space, not all in a plane
Johnson_solid
On least area of curves of constant width
who published it separately in the early 20th century. The width of a convex set K {\displaystyle K} in the Euclidean plane is defined as the minimum distance
Blaschke–Lebesgue_theorem
Class of algorithms in computational geometry
proposed for computing the convex hull of a finite set of points, with various computational complexities. Computing the convex hull means that a non-ambiguous
Convex_hull_algorithms
Construct in functional analysis
hull of a convex set may fail to be convex (however, the convex hull of a balanced set is always balanced). For an example, let the convex subset be S
Balanced_set
Black-box description of a convex set
oracle) is a concept in the mathematical theory of convex optimization. It is a method to describe a convex set that is given as an input to an optimization
Separation_oracle
Sequences of convex sets in a bounded set have convergent subsequences
topology and convex geometry about sequences of convex sets. Specifically, given a sequence { K n } {\displaystyle \{K_{n}\}} of convex sets contained in
Blaschke_selection_theorem
Biconvex optimization is a generalization of convex optimization where the objective function and the constraint set can be biconvex. There are methods that
Biconvex_optimization
Generalization of topological interior
general sets. They are equal if both S 1 , S 2 {\displaystyle S_{1},S_{2}} are also convex. If S 1 , S 2 {\displaystyle S_{1},S_{2}} are convex and relatively
Relative_interior
Theorem about the intersections of d-dimensional convex sets
theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913, but not published by him
Helly's_theorem
Directed graph with no directed cycles
hull of a subset of vertices S is the smallest convex set containing S. That is, the convex hull of a set S in a DAG consists of all vertices that lie on
Directed_acyclic_graph
is: If K is any bounded convex set in the n-dimensional Euclidean space Rn, then there exists a set of 2n scalars si and a set of 2n translation vectors
Hadwiger conjecture (combinatorial geometry)
Hadwiger_conjecture_(combinatorial_geometry)
Inequality which involves a linear function
inequalities. It is a convex set, since the half-spaces are convex sets, and the intersection of a set of convex sets is also convex. In the non-degenerate
Linear_inequality
approximately Euclidean. Equivalently, every high-dimensional bounded symmetric convex set has low-dimensional sections that are approximately ellipsoids. A new
Dvoretzky's_theorem
Generalization of the concept of a norm
Different convex sets yield different seminorms, and every asymmetric seminorm on R n {\displaystyle \mathbb {R} ^{n}} can be obtained from some convex set, called
Asymmetric_norm
Vector space with a notion of nearness
intersection of any family of convex sets is convex and the convex hull of a subset is equal to the intersection of all convex sets that contain it. Properties
Topological_vector_space
Sums of sets of vectors are nearly convex
The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. The lemma may be intuitively
Shapley–Folkman_lemma
Five coplanar points have a subset forming a convex quadrilateral
statement: Theorem—any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral. This
Happy_ending_problem
Distance function
distance in the Cayley–Klein model of hyperbolic geometry, where the convex set is the n-dimensional open unit ball. Hilbert's metric has been applied
Hilbert_metric
Shape with same width in all directions
circles centered on a partial curve. Every body of constant width is a convex set, its boundary crossed at most twice by any line, and if the line crosses
Curve_of_constant_width
Mathematical embedding in Banach spaces
(x)(y)=d(x,y)\quad {\mbox{for all}}\quad x,y\in X} The convex set mentioned above is the convex hull of Ψ(X). In both of these embedding theorems, we may
Kuratowski_embedding
Correspondence in functional analysis
\{0\}} . Theorem—The set of states of a C ∗ {\displaystyle C^{*}} -algebra A {\displaystyle A} with a unit element is a compact convex set under the weak-
Gelfand–Naimark–Segal construction
Gelfand–Naimark–Segal_construction
Theorems generalizing the Brouwer fixed-point theorem
fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : X → X
Fixed-point theorems in infinite-dimensional spaces
Fixed-point_theorems_in_infinite-dimensional_spaces
Measure of difference between two points
\mathbb {R} } be a continuously-differentiable, strictly convex function defined on a convex set Ω {\displaystyle \Omega } . The Bregman distance associated
Bregman_divergence
On the existence of hyperplanes separating disjoint convex sets
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
Hyperplane_separation_theorem
Geometric shape
self-intersecting torus). The lemon forms the boundary of a convex set, while its surrounding apple is non-convex. The ball in North American football has a shape
Lemon_(geometry)
Periodic set of points
\mathrm {d} (\Lambda )} , or more generally the volume of a symmetric convex set S {\displaystyle S} , to the number of lattice points contained in S
Lattice_(group)
Infinite sum of monomials
region, is a convex set. More generally, one can show that when c=0, the interior of the region of absolute convergence is always a log-convex set in this
Power_series
Property of group subsets (mathematics)
symmetric sets. Any balanced subset of a real or complex vector space is symmetric. Absolutely convex set – Convex and balanced set Absorbing set – Set that
Symmetric_set
Set of probability measures
measures. A credal set is often assumed or constructed to be a closed convex set. It is intended to express uncertainty or doubt about the probability
Credal_set
Graph with a median for each three vertices
S} . Observe that the intersection of every pair of convex sets is itself convex. The convex sets in a median graph have the Helly property: if F {\displaystyle
Median_graph
Area of functional analysis and convex analysis
of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every
Choquet_theory
In computational geometry, the convex layers of a set of points in the Euclidean plane are a sequence of nested convex polygons having the points as their
Convex_layers
Theorem in geometry about convex sets
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 sets whose convex hulls intersect
Radon's_theorem
Sets whose elements have degrees of membership
fuzzy set A ( U ⊆ R ) {\displaystyle A(U\subseteq \mathbb {R} )} is said to be convex (in the fuzzy sense, not to be confused with a crisp convex set), iff
Fuzzy_set
Type of mathematical function
In convex analysis, a non-negative function f: Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it
Logarithmically concave function
Logarithmically_concave_function
Subset of all points that is bounded by some given point of a dual (in a dual pairing)
functional and convex analysis, and related disciplines of mathematics, the polar set A ∘ {\displaystyle A^{\circ }} is a special convex set associated to
Polar_set
Set of vectors in convex analysis
In mathematics, especially convex analysis, the recession cone of a set A {\displaystyle A} is a cone containing all vectors such that A {\displaystyle
Recession_cone
Shape that can be represented as a linear matrix inequality
In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of n × n positive semidefinite
Spectrahedron
Type of function in linear algebra
: X → R {\displaystyle p\colon X\to \mathbb {R} } which is subadditive, convex, and satisfies p ( 0 ) ≤ 0 {\displaystyle p(0)\leq 0} is also positively
Sublinear_function
is compact), but it is most useful for convex bodies (that is bodies, whose corresponding set is a convex set). The mean width of a line segment L is
Mean_width
Largest distance between two points
Euclidean space, the diameter of the object or set is the same as the diameter of its convex hull. For any convex shape in the plane, the diameter is the largest
Diameter_of_a_set
to the definition of a convex set in geometry, a set that contains the line segment between every pair of its points. Convex subgraphs play an important
Convex_subgraph
Mathematical concept
Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex. When G {\displaystyle G}
Pseudoconvexity
Bisection of Euclidean space by a hyperplane
hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any subspace connecting a point in one set to a point in the other must intersect
Half-space_(geometry)
Impossibility result for ranked-choice voting systems
extends the results of Border and Jordan by allowing the set of candidates M to be any convex subset of Rd. An example scenario is voting on the amount
Gibbard–Satterthwaite_theorem
travel, tourism, insurance
CONVEX SET
CONVEX SET
CONVEX SET
CONVEX SET
CONVEX SET
CONVEX SET
CONVEX SET
CONVEX SET
CONVEX SET
travel, tourism, insurance