Affine cone may refer to:
In geometry, a set of points is convex if it contains every line segment between two points in the set. Equivalently, a convex set or a convex region is a set that intersects every line in a line segment, single point, or the empty set. 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.
In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results. The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.
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In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special, but still quite general class of toric varieties, this information is also encoded in a polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space.
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.
In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities.
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.
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.
In mathematics, the affine hull or affine span of a set S in Euclidean space Rn is the smallest affine set containing S, or equivalently, the intersection of all affine sets containing S. Here, an affine set may be defined as the translation of a vector subspace.
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, is a cone if implies for every positive scalar . 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.
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.
A second-order cone program (SOCP) is a convex optimization problem of the form
In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces.
Conic optimization is a subfield of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone.
In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality:
Given a finite number of vectors in a real vector space, a conical combination, conical sum, or weighted sum of these vectors is a vector of the form
In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the body and is invariant under linear transformations. It is named after German-English mathematician Kurt Mahler. It is known that the shapes with the largest possible Mahler volume are the balls and solid ellipsoids; this is now known as the Blaschke–Santaló inequality. The still-unsolved Mahler conjecture states that the minimum possible Mahler volume is attained by a hypercube.
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 matrices forms a convex cone in Rn × n, and a spectrahedron is a shape that can be formed by intersecting this cone with an affine subspace.
In abstract algebra, a branch of mathematics, an affine monoid is a commutative monoid that is finitely generated, and is isomorphic to a submonoid of a free abelian group . Affine monoids are closely connected to convex polyhedra, and their associated algebras are of much use in the algebraic study of these geometric objects.