Convex space

Last updated February 03, 2024

In mathematics, a convex space (or barycentric algebra) is a space in which it is possible to take convex combinations of any sets of points.^[1]^[2]

Formal Definition

A convex space can be defined as a set $X$ equipped with a binary convex combination operation $c_{\lambda }:X\times X\rightarrow X$ for each $\lambda \in [0,1]$ satisfying:

$c_{0}(x,y)=x$
$c_{1}(x,y)=y$
$c_{\lambda }(x,x)=x$
$c_{\lambda }(x,y)=c_{1-\lambda }(y,x)$
$c_{\lambda }(x,c_{\mu }(y,z))=c_{\lambda \mu }\left(c_{\frac {\lambda (1-\mu )}{1-\lambda \mu }}(x,y),z\right)$ (for $\lambda \mu \neq 1$ )

From this, it is possible to define an n-ary convex combination operation, parametrised by an n-tuple $(\lambda _{1},\dots ,\lambda _{n})$ , where $\sum _{i}\lambda _{i}=1$ .

Examples

Any real affine space is a convex space. More generally, any convex subset of a real affine space is a convex space.

History

Convex spaces have been independently invented many times and given different names, dating back at least to Stone (1949).^[3] They were also studied by Neumann (1970)^[4] and Świrszcz (1974),^[5] among others.

Related Research Articles

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if

In mathematics, the $L p$ spaces are function spaces defined using a natural generalization of the $p$ -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz.

In mathematics, the linear span (also called the linear hull or just span) of a set $S$ of vectors (from a vector space), denoted $span(S)$ , is defined as the set of all linear combinations of the vectors in $S$ . For example, two linearly independent vectors span a plane. The linear span can be characterized either as the intersection of all linear subspaces that contain $S$ , or as the smallest subspace containing $S$ . The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules.

<span class="mw-page-title-main">Jensen's inequality</span> Theorem of convex functions

In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. Affine space is the setting for affine geometry.

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:

In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".

In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex. The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex.

In mathematics and functional analysis, a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series On Rings of Operators. One of von Neumann's goals in this paper was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings.

Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.

In mathematics, the Brunn–Minkowski theorem is an inequality relating the volumes of compact subsets of Euclidean space. The original version of the Brunn–Minkowski theorem applied to convex sets; the generalization to compact nonconvex sets stated here is due to Lazar Lyusternik (1935).

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.

In mathematics, a Borel measure μ on n-dimensional Euclidean space $is called logarithmically concave if, for any compact subsets A and B of and 0 < λ < 1, one has$

In mathematics, in particular in measure theory, a content $is a real-valued function defined on a collection of subsets such that$

In mathematics, Macdonald polynomialsP_λ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald originally associated his polynomials with weights λ of finite root systems and used just one variable t, but later realized that it is more natural to associate them with affine root systems rather than finite root systems, in which case the variable t can be replaced by several different variables t=(t₁,...,t_k), one for each of the k orbits of roots in the affine root system. The Macdonald polynomials are polynomials in n variables x=(x₁,...,x_n), where n is the rank of the affine root system. They generalize many other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials and Askey–Wilson polynomials, which in turn include most of the named 1-variable orthogonal polynomials as special cases. Koornwinder polynomials are Macdonald polynomials of certain non-reduced root systems. They have deep relationships with affine Hecke algebras and Hilbert schemes, which were used to prove several conjectures made by Macdonald about them.

In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In differential geometry, a fibered manifold is surjective submersion of smooth manifolds $Y \to X$ . Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles.

In measure and probability theory in mathematics, a convex measure is a probability measure that — loosely put — does not assign more mass to any intermediate set "between" two measurable sets A and B than it does to A or B individually. There are multiple ways in which the comparison between the probabilities of A and B and the intermediate set can be made, leading to multiple definitions of convexity, such as log-concavity, harmonic convexity, and so on. The mathematician Christer Borell was a pioneer of the detailed study of convex measures on locally convex spaces in the 1970s.

In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced by Gang Tian and reformulated more algebraically later by Simon Donaldson. The definition was inspired by a comparison to geometric invariant theory (GIT) stability. In the special case of Fano varieties, K-stability precisely characterises the existence of Kähler–Einstein metrics. More generally, on any compact complex manifold, K-stability is conjectured to be equivalent to the existence of constant scalar curvature Kähler metrics.

References

↑ "Convex space". nLab. Retrieved 3 April 2023.
↑ Fritz, Tobias (2009). "Convex Spaces I: Definition and Examples". arXiv: 0903.5522 [math.MG].
↑ Stone, Marshall Harvey (1949). "Postulates for the barycentric calculus". Annali di Matematica Pura ed Applicata. 29: 25–30. doi:10.1007/BF02413910. S2CID 122252152.
↑ Neumann, Walter David (1970). "On the quasivariety of convex subsets of affine spaces". Archiv der Mathematik. 21: 11–16. doi:10.1007/BF01220869. S2CID 124051153.
↑ Świrszcz, Tadeusz (1974). "Monadic functors and convexity". Bulletin l'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques. 22: 39–42.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] "Convex space". nLab. Retrieved 3 April 2023.

[2] Fritz, Tobias (2009). "Convex Spaces I: Definition and Examples". arXiv: 0903.5522 [math.MG].

[3] Stone, Marshall Harvey (1949). "Postulates for the barycentric calculus". Annali di Matematica Pura ed Applicata. 29: 25–30. doi:10.1007/BF02413910. S2CID 122252152.

[4] Neumann, Walter David (1970). "On the quasivariety of convex subsets of affine spaces". Archiv der Mathematik. 21: 11–16. doi:10.1007/BF01220869. S2CID 124051153.

[5] Świrszcz, Tadeusz (1974). "Monadic functors and convexity". Bulletin l'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques. 22: 39–42.

[1]

[2]

[3]

[4]

[5]