convex space

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.{{cite web |title=Convex space |url=https://ncatlab.org/nlab/show/convex+space |website=nLab |access-date=3 April 2023}}{{cite arXiv |last1=Fritz |first1=Tobias |title=Convex Spaces I: Definition and Examples |year=2009 |class=math.MG |eprint=0903.5522 }}

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).{{cite journal |last1=Stone |first1=Marshall Harvey |title=Postulates for the barycentric calculus |journal=Annali di Matematica Pura ed Applicata |date=1949 |volume=29 |pages=25–30|doi=10.1007/BF02413910 |s2cid=122252152 }} They were also studied by Neumann (1970){{cite journal |last1=Neumann |first1=Walter David |title=On the quasivariety of convex subsets of affine spaces |journal=Archiv der Mathematik |date=1970 |volume=21 |pages=11–16|doi=10.1007/BF01220869 |s2cid=124051153 }} and Świrszcz (1974),{{cite journal |last1=Świrszcz |first1=Tadeusz |title=Monadic functors and convexity |journal=Bulletin l'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques |date=1974 |volume=22 |pages=39–42}} among others.

References

Category:Convex geometry

Category:Algebraic structures