List of convexity topics

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This is a list of convexity topics, by Wikipedia page.

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<span class="mw-page-title-main">Convex set</span> In geometry, set whose intersection with every line is a single line segment

In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment . 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.

<span class="mw-page-title-main">Simplex</span> Multi-dimensional generalization of triangle

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example,

<span class="mw-page-title-main">Convex hull</span> 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 either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.

<span class="mw-page-title-main">Hyperplane</span> Subspace of n-space whose dimension is (n-1)

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line.

<span class="mw-page-title-main">Geometry of numbers</span>

Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in and the study of these lattices provides fundamental information on algebraic numbers. The geometry of numbers was initiated by Hermann Minkowski.

<span class="mw-page-title-main">Discrete geometry</span> Branch of geometry that studies combinatorial properties and constructive methods

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

<span class="mw-page-title-main">Minkowski addition</span> Sums vector sets A and B by adding each vector in A to each vector in B

In geometry, the Minkowski sum of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B:

<span class="mw-page-title-main">Convex polygon</span> 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 contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon. Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points.

<span class="mw-page-title-main">Radon's theorem</span> Says d+2 points in d dimensions can be partitioned into two subsets whose convex hulls intersect

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 sets whose convex hulls intersect.

<span class="mw-page-title-main">Real coordinate space</span> Space formed by the n-tuples of real numbers

In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted Rn or , is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors. Special cases are called the real lineR1, the real coordinate planeR2, and the real coordinate three-dimensional spaceR3. With component-wise addition and scalar multiplication, it is a real vector space.

<span class="mw-page-title-main">Convex polytope</span> 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 -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.

<span class="mw-page-title-main">Convex cone</span> Mathematical set closed under positive linear combinations

In linear algebra, a cone—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under positive scalar multiplication; that is, C is a cone if implies for every positive scalar s. A cone need not be convex, or even look like a cone in Euclidean space.

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. Here C is a subset of a real vector space V, and the main thrust of the theory is to treat the cases where V is an infinite-dimensional topological vector space along lines similar to the finite-dimensional case. The main concerns of Gustave Choquet were in potential theory. Choquet theory has become a general paradigm, particularly for treating convex cones as determined by their extreme rays, and so for many different notions of positivity in mathematics.

<span class="mw-page-title-main">Supporting hyperplane</span>

In geometry, a supporting hyperplane of a set in Euclidean space is a hyperplane that has both of the following two properties:

In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, game theory, etc.

<span class="mw-page-title-main">Hyperplane separation theorem</span> 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 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.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.

<span class="mw-page-title-main">Integral polytope</span> A convex polytope whose vertices all have integer Cartesian coordinates

In geometry and polyhedral combinatorics, an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex hull of its integer points. Integral polytopes are also called lattice polytopes or Z-polytopes. The special cases of two- and three-dimensional integral polytopes may be called polygons or polyhedra instead of polytopes, respectively.

In geometry, the moment curve is an algebraic curve in d-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form

In geometry, it is an unsolved conjecture of Hugo Hadwiger that every simplex can be dissected into orthoschemes, using a number of orthoschemes bounded by a function of the dimension of the simplex. If true, then more generally every convex polytope could be dissected into orthoschemes.