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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:
Convex_geometry
In geometry, set whose intersection with every line is a single line segment
In 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
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
Convex polyhedron with regular faces
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons and that is not
Johnson_solid
Russian mathematician (born 1966)
in the field of convex geometry. His first published article studied the combinatorial structures arising from intersections of convex polyhedra.[P85]
Grigori_Perelman
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
Type of plane curve
In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these
Convex_curve
Branch of mathematics
groups are sometimes regarded as strongly geometric as well. Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues
Geometry
Mathematical set closed under positive linear combinations
(disambiguation) Cone (geometry) Cone (topology) Farkas' lemma Bipolar theorem Ordered vector space Boyd, Stephen; Vandenberghe, Lieven (2004-03-08). Convex Optimization
Convex_cone
Planar surface that forms part of the boundary of a solid object
Discrete Geometry, Graduate Texts in Mathematics, vol. 212, Springer, ISBN 9780387953748, MR 1899299 Rockafellar, R. T. (1997) [1970]. Convex Analysis
Face_(geometry)
Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P
Carathéodory's theorem is a theorem in convex geometry. It states that if a point x {\displaystyle x} lies in the convex hull C o n v ( P ) {\displaystyle
Carathéodory's theorem (convex hull)
Carathéodory's_theorem_(convex_hull)
Application of geometry in number theory
and lattice theory. The geometry of numbers contributed to the development of convex geometry. A central operation on convex bodies is the Minkowski sum
Geometry_of_numbers
Non-empty convex set in Euclidean space
contained in, an n-dimensional convex object Brunn–Minkowski theorem, which has many implications relevant to the geometry of convex bodies. Hug, Daniel; Weil
Convex_body
Convex hull of a finite set of points in a Euclidean space
Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out
Convex_polytope
Flat-sided three-dimensional shape
In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many' and ἕδρον (-hedron) 'base, seat') is a three-dimensional figure
Polyhedron
Type of metric space in mathematics
of "convexity" on metric spaces. Karl Menger defined a metric space as convex if any "segment" joining two points in that space has other points in it
Convex_metric_space
Branch of computer science
Computational geometry is a branch of computer science devoted to the study of algorithms that can be stated in terms of geometry. Some purely geometrical
Computational_geometry
Branch of geometry that studies combinatorial properties and constructive methods
Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial
Discrete_geometry
Linear combination of points
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points
Convex_combination
Overview of and topical guide to geometry
solid geometry Contact geometry Convex geometry Descriptive geometry Differential geometry Digital geometry Discrete geometry Distance geometry Elliptic
Outline_of_geometry
Topics referred to by the same term
affine geometry Conical hull, in convex geometry Convex hull, in convex geometry Carathéodory's theorem (convex hull) Holomorphically convex hull, in
Hull
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
Algebraic variety containing an algebraic torus
information is also encoded in a convex polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties
Toric_variety
In discrete geometry, a polytope is projectively unique (or projectively stable) if it has a unique convex realization up to projective transformations
Projectively_unique_polytope
Structure in convex geometry
In mathematics, specifically convex geometry, the normal fan of a convex polytope P is a polyhedral fan that is dual to P. Normal fans have applications
Normal_fan
Generalization of the tangent space to a manifold to the case of certain spaces
ISBN 978-0-8176-4848-0. A. D. Aleksandrov (2006). Intrinsic geometry of convex surfaces. Chapman & Hall/CRC Press. Chapman & Hall/CRC Press. doi:10
Tangent_cone
Distance from origin of tangent hyperplanes
in convex geometry. The support function h A : R n → R {\displaystyle h_{A}\colon \mathbb {R} ^{n}\to \mathbb {R} } of a non-empty closed convex set
Support_function
Mathematics of convex functions and sets
convex geometry, economics, and related fields. A set is convex if it contains every line segment joining two of its points. A function is convex if
Convex_analysis
Greek mathematician (1873–1950)
Carathéodory's theorem in convex geometry states that if a point x {\displaystyle x} of R d {\displaystyle \mathbb {R} ^{d}} lies in the convex hull of a set P
Constantin_Carathéodory
German mathematician (1905–1988)
mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization
Werner_Fenchel
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
Convex plane region bounded by two circular arcs
2-dimensional geometry, a lens is a convex region bounded by two circular arcs joined to each other at their endpoints. In order for this shape to be convex, both
Lens_(geometry)
Branch of mathematics
objects, such as convex bodies and normed spaces, as the dimension tends to infinity. It is at the intersection of convex geometry and functional analysis
Asymptotic_geometry
Measure method in computational geometry
images Convex polygon Convex hull Smallest enclosing box "Rotating Calipers" at Toussaint's home page Shamos, Michael (1978). "Computational Geometry" (PDF)
Rotating_calipers
A convex cap is a well defined structure in mathematics commonly used in convex geometry for approximating convex shapes. It is used in the construction
Convex_cap
Hyperplane in geometry
In geometry, a supporting hyperplane of a set S {\displaystyle S} in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is a hyperplane that has both
Supporting_hyperplane
On the existence of hyperplanes separating disjoint convex sets
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar
Hyperplane_separation_theorem
Theorem in geometry about convex sets
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
Radon's_theorem
Class of algorithms in computational geometry
construct convex hulls of various objects have a broad range of applications in mathematics and computer science. In computational geometry, numerous
Convex_hull_algorithms
Raphael (2020), "Topological drawings meet classical theorems from convex geometry", Proceedings of the 28th International Symposium on Graph Drawing
Kirchberger's_theorem
onto 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
German mathematician and physicist (1864–1909)
Lithuanian-German, or Russian. He created and developed the geometry of numbers and elements of convex geometry, and used geometrical methods to solve problems in
Hermann_Minkowski
Point where two or more curves, lines, or edges meet
In geometry, a vertex (pl.: vertices or vertexes), also called a corner, is a point where two or more curves, lines, or line segments meet or intersect
Vertex_(geometry)
Doignon's theorem in geometry is an analogue of Helly's theorem for the integer lattice. It states that, if a family of convex sets in d {\displaystyle
Doignon's_theorem
Theorem about the intersections of d-dimensional convex sets
Helly's 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
Helly's_theorem
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)
Rigidity theorem for convex polyhedra
Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states that convex polytopes in three dimensions with congruent corresponding
Cauchy's_theorem_(geometry)
Concepts in convex analysis
Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics. The dual cone C* of a subset C in a linear space X over
Dual_cone_and_polar_cone
Mathematical subject
faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which
Geometric_combinatorics
Polytope combining two smaller polytopes
In convex geometry and the geometry of convex polytopes, the Blaschke sum of two polytopes is a polytope that has a facet parallel to each facet of the
Blaschke_sum
Polyhedra are determined by surface distance
describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes
Alexandrov's theorem on polyhedra
Alexandrov's_theorem_on_polyhedra
Four-dimensional analogues of the regular polyhedra in three dimensions
Still "Convex and abstract polytopes", Programme and abstracts, MIT, 2005 Johnson, Norman W. (2018). "§ 11.5 Spherical Coxeter groups". Geometries and Transformations
Regular_4-polytope
Open problem in convex geometry
In mathematics, the affine plank problem is an open question in convex geometry posed by Thøger Bang in 1951 as a strengthening of Tarski's plank problem
Affine_plank_problem
theorem (discrete geometry) Busemann's theorem (Euclidean geometry) Carathéodory's theorem (convex geometry) Cauchy's theorem (geometry) Classification
List_of_theorems
manifold. Convex analysis the study of properties of convex functions and convex sets. Convex geometry part of geometry devoted to the study of convex sets
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Skeletonized version of algebraic geometry
Convex Geometry as the Ricardian Theory of International Trade" draft paper. Zhang, Liwen; Naitzat, Gregory; Lim, Lek-Heng (2018). "Tropical Geometry
Tropical_geometry
Method of determining minimum distance between two convex sets
distance algorithm is a method of determining the minimum distance between two convex sets, first published by Elmer G. Gilbert, Daniel W. Johnson, and S. Sathiya
Gilbert–Johnson–Keerthi distance algorithm
Gilbert–Johnson–Keerthi_distance_algorithm
Mathematical theorem
mathematical theorem in the fields of mathematical statistics and convex geometry. The Gaussian correlation inequality states: Let μ {\displaystyle \mu
Gaussian correlation inequality
Gaussian_correlation_inequality
Shape with three sides
three sides connected at three corners. It is one of the basic shapes in geometry and the simplest of polygons. The corners, also called vertices, are zero-dimensional
Triangle
Sums vector sets A and B by adding each vector in A to each vector in B
Discrete & Computational Geometry, 35 (2): 223–240, doi:10.1007/s00454-005-1206-y Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton landmarks
Minkowski_addition
Convex and balanced set
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
Absolutely_convex_set
American mathematician
York University in New York City. His main research interests are convex geometry and its connections with analysis and information theory. Gaoyong Zhang
Gaoyong_Zhang
Part of a straight line that is bounded by two distinct end points
is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points. In geometry, one might
Line_segment
In functional analysis, the class of B-convex spaces is a class of Banach space. The concept of B-convexity was defined and used to characterize Banach
B-convex_space
Quadrilateral symmetric across a diagonal
In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and
Kite_(geometry)
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
Dykstra's projection algorithm
Dykstra's_projection_algorithm
Straight figure with zero width and depth
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature. It is a special case of a curve
Line_(geometry)
Theorem in geometry
much insight into the geometry of high dimensional convex bodies. In this section we sketch a few of those insights. Consider a convex body K ⊆ R n {\textstyle
Brunn–Minkowski_theorem
Theorem in convex and algebraic geometry
Gordan's lemma is a lemma in convex geometry and algebraic geometry. It can be stated in several ways. Let A {\displaystyle A} be a matrix of integers
Gordan's_lemma
Theorem in topology
any continuous function f {\displaystyle f} mapping a nonempty compact convex set to itself, there is a point x 0 {\displaystyle x_{0}} such that f (
Brouwer_fixed-point_theorem
the k {\displaystyle k} -skeleton of the polytope. The edge graph of a convex polytope is a finite simple graph. It is connected, since a path between
Graph_of_a_polytope
Fixed-point theorem for set-valued functions
It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point
Kakutani_fixed-point_theorem
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
In convex geometry, the projection body Π K {\displaystyle \Pi K} of a convex body K {\displaystyle K} in n-dimensional Euclidean space is the convex body
Projection_body
Minkowsi sum of line segments
A zonotope is a convex polytope that can be described as the Minkowski sum of a finite set of line segments in R d {\displaystyle \mathbb {R} ^{d}} or
Zonotope
Significant topic in economics
economics depends upon the following definitions and results from convex geometry. A real vector space of two dimensions may be given a Cartesian coordinate
Convexity_in_economics
Russian mathematician (born 1936)
June 1936) is a Russian mathematician. He works in differential and convex geometry. Burago studied at Leningrad University, where he obtained his Ph.D
Yuri_Burago
Type of non-Euclidean geometry
mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate
Hyperbolic_geometry
German mathematician
University of Freiburg. His main research interests are convex geometry and stochastic geometry. Schneider completed his PhD 1967 with Ruth Moufang at
Rolf_Schneider
Subspace of n-space whose dimension is (n-1)
is generated by the reflections. A convex polytope is the intersection of half-spaces. In non-Euclidean geometry, the ambient space might be the n-dimensional
Hyperplane
Convex quadrilateral with at least one pair of parallel sides
usually considered to be a convex quadrilateral in Euclidean geometry, but there are also crossed cases. If shape ABCD is a convex trapezoid, then the ABDC
Trapezoid
Class of convex shapes
In convex geometry, a zonoid is a type of centrally symmetric convex body. The zonoids have several definitions, equivalent up to translations of the
Zonoid
Kakutani's theorem is a result in geometry named after Shizuo Kakutani. It states that every convex body in 3-dimensional space has a circumscribed cube
Kakutani's_theorem_(geometry)
On partitions into intersecting convex hulls
In discrete geometry, Tverberg's theorem, first stated by Helge Tverberg in 1966, is the result that sufficiently many points in Euclidean space can be
Tverberg's_theorem
In mathematics, Busemann's theorem is a theorem in Euclidean geometry and geometric tomography. It was first proved by Herbert Busemann in 1949 and was
Busemann's_theorem
Mathematical model of the physical space
Euclidean geometry is a mathematical system attributed to Euclid, an ancient Greek mathematician, which he described in his textbook on geometry, Elements
Euclidean_geometry
1967 mathematics textbook
and convex geometry, two more chapters provide basic definitions of polyhedra, in their two dual versions (intersections of half-spaces and convex hulls
Convex_Polytopes
In mathematics, most commonly in convex geometry, an extreme set or face of a set C ⊆ V {\displaystyle C\subseteq V} in a vector space V {\displaystyle
Extreme_set
Branch of discrete mathematics
discrete geometry. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study
Combinatorics
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
Blaschke_selection_theorem
Mexican mathematician
whose research interests include discrete geometry, combinatorics, and convex geometry, including the geometry of bodies of constant width and related topics
Déborah_Oliveros
Ellipsoid most closely containing, or contained in, an n-dimensional convex object
mathematics, the John ellipsoid or Löwner–John ellipsoid E(K) associated to a convex body K in n-dimensional Euclidean space R n {\displaystyle \mathbb {R}
John_ellipsoid
Unit hypercube of variable dimension whose corners have been perturbed
Sequential quadratic programming Successive linear programming Convex optimization Convex minimization Cutting-plane method Reduced gradient (Frank–Wolfe)
Klee–Minty_cube
Archimedean solid with 62 faces
In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of
Rhombicosidodecahedron
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
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
German American mathematician
geometry. Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-5198-2. Sturmfels, Bernd (1998). "Polynomial equations and convex polytopes"
Bernd_Sturmfels
Line segment joining two adjacent vertices in a polygon or polytope
edges of a 3-dimensional convex polyhedron are its ridges, and the edges of a 4-dimensional polytope are its peaks. Base (geometry) Extended side Ziegler
Edge_(geometry)
Problem in convex geometry
In the mathematical field of convex geometry, the Busemann–Petty problem, introduced by Herbert Busemann and Clinton Myers Petty (1956, problem 1), asks
Busemann–Petty_problem
Graph whose biconnected components are all cliques
the connected subsets of vertices in a connected block graph form a convex geometry, a property that is not true of any graphs that are not block graphs
Block_graph
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CONVEX GEOMETRY
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