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\begin{document}

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\section*{convex space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{analysis}{}\paragraph*{{Analysis}}\label{analysis}

[[!include analysis - contents]]

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{abstract_convex_sets}{Abstract convex sets}\dotfill \pageref*{abstract_convex_sets} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{convex space} (also called \textbf{barycentric algebra} and other terms, invented independently many times) is a [[set]] [[extra structure|equipped]] with a notion of taking weighted averages, or convex-[[linear combinations]], of its elements. Do not confuse this with an (abstract) \emph{[[convex set]]} , which a special kind of convex space, also defined below.

The [[category]] of convex spaces is an algebraic theory, being the affine part of the theory of $K$-(semi)modules with only the idempotent operations. This definition is used by \hyperlink{Meng}{Meng (1989)}, and many basic properties of the category are detailed therein. The category is complete, cocomplete, symmetric monoidal closed under the (usual) tensor product construction, and has a cogenerator \hyperlink{Borger}{Borger and Kemper (1994)}. The subcategory consisting of the single object, the unit interval, is dense (left-adequate) in the category. This follows from Isbell's theorem on left adequate subcategories for algebraic theories, using the fact that the free convex space on 2 elements is the unit interval.

Axiomatically, a convex space can be characterized as a [[set]] $X$ equipped with a family of [[functions]] $c_p : X \times X \to X$ satisfying some natural [[axioms]] (described below).

For examples All [[nonunital ring|nonunital]] [[commutative rings]] are convex spaces, with the map $c_p(x,y) = x + p(y-x)$.

The [[monad]] assigning to any set the free convex space on that set is a [[finitary monad|finitary]] [[commutative monad]]. We can thus follow Durov in thinking of it as a [[generalized ring]]. This allows us to think of convex spaces as `modules' of a generalized ring, very much as [[vector spaces]] are modules of a field. This is also true of the relatives of convex spaces: [[affine space|affine spaces]] and [[conical space|conical spaces]]. For example, all \textbf{affine spaces} are convex spaces as defined below.

Of particular importance are convex spaces parametrized by the interval $P = [0,1]$ or the Boolean algebra $P = \{0,1\}$. These two algebras are dual, in a certain sense described by \hyperlink{Jacobs}{Jacobs (2009)}. This duality is functorial, and therefore is present for convex spaces for general $P$. This leads to the notion of a [[dual convex space]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{convex space} is a [[set]] $X$ equipped with:

\begin{itemize}%
\item a multiplicatively closed [[subset]] $Q$ of a [[semiring|(semi)]][[ring]] $P$, so that for each element $p\in Q$ there exists an element $q\in Q$ such that $p+q=1$, and


\item an operation $c_p: X \times X \to X$ defined for all $p\in Q$,



\end{itemize}
such that the following identities always hold:

\begin{itemize}%
\item $c_0(x,y) = x$,
\item $c_p(x,x) = x$ for all $p \in P$,
\item $c_p(x,y) = c_{1-p}(y,x)$ for all $p \in P$ (in semiring case replace $1-p$ by $q$ and require that $p+q = 1$),
\item $c_p(x, c_q(y,z)) = c_{p q}(c_r(x,y),z)$ for all $p,q,r \in P$ satisfying $p(1 - q) = (1 - p q)r$.

\end{itemize}
As a consequence of the first and third axioms, $c_1(x,y) = c_0(y,x) = y$.

This defines convex spaces as a [[variety of algebras]], with one binary operation for each $p$.

The intended interpretation is that $c_p(x,y) = x + p(y-x) = (1-p)x + p y$. i.e., $c_p(x,y)$ is the $p$-weighted average of $x$ and $y$, where $x$ gets weight $1-p$ and $y$ gets weight $p$. By thinking of $p$ as a continuous parameter, this interpretation has the advantage of ``starting'' at $x$, then moving toward $y$ at ``rate'' $p$.

This interpretation is `[[bias|biased]]', in the sense that the centered choice $p=0$ favors $x$. It is also possible to give an `[[bias|unbiased]]' definition, which characterizes to convex-linear combinations of many points. This is an $n$-ary operation parametrised by a list $p := (p_1,\ldots,p_n)$ satisfying $\sum_{i = 1}^n p_i = 1$. If $x := (x^1,\ldots, x^n)$, then $c_p(x) := \sum_i p_i x^i$.

A [[homomorphism]] of convex spaces may be called a \textbf{convex-linear map} or an \textbf{affine linear map} (since an [[affine space]] is a convex space with extra properties, as in the examples below). It should probably \emph{not} be called a `convex map', which (between affine spaces) is something more general.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Any real [[vector space]] is a convex space, with $c_p(x,y) = x + p(y-x)$. In the unbiased version, any convex-linear combination is a [[linear combination]]. Note that a convex-linear map between vector spaces may not be a linear map, since it may not preserve the identity; thus, a vector space is a convex space with [[extra structure]].

More generally, any real [[affine space]] is a convex space; since $p + (1 - p) = 1$, the expression for $c_p$ in a vector space is valid in an affine space. In the unbiased version, any convex-linear combination is an [[affine linear combination]]. Now any convex-linear map between affine spaces is an affine linear map (and conversely); an affine space is a convex space with [[extra properties]].

Still more generally, any [[convex subset]] (that is, one containing the entire line segment between two given points) of a real [[affine space]] is a convex space (again with extra properties, which are described algebraically below).

The [[Boolean field]] $\{0,1\}$ is a convex space with $c_p(x,y) = x \vee y = x + y - x y$ whenever $0 \lt p \lt 1$ (with $c_0(x,y) = x$ and $c_1(x,y) = y$ as always); this cannot be realised as a subset of a vector space. This can be generalised to any (possibly unbounded) [[semilattice]]. (It would be nice to find an example like this that can be defined constructively; this one relies on [[excluded middle]].)

\hypertarget{abstract_convex_sets}{}\subsection*{{Abstract convex sets}}\label{abstract_convex_sets}

There is a nice abstract converse to the example of a [[convex subset]] of an affine space. A convex space is \textbf{cancellative} if $y = z$ whenever $c_p(x,y) = c_p(x,z)$ for some $c$ and $p \ne 0$. We may call a cancellative convex space an \textbf{abstract convex set}. The justification for this terminology is this

\begin{utheorem}
A convex space is cancellative if and only if it is isomorphic (as a convex space) to a convex subset of some real affine space.

\end{utheorem}
Compare this with the theorem that a [[monoid]] is [[cancellative monoid|cancellative]] if and only if it is isomorphic to a submonoid of some [[group]].

Of course, most of the examples given above are cancellative, being manifestly given as convex subsets of real affine space. However, the last example --- a semilattice with $c_p(x,y) = x \vee y$ whenever $0 \lt p \lt 1$ --- is non-cancellative.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[convex function]]
\item [[distribution monad]]
\item [[monads of probability, measures, and valuations]]
\item [[vector space]], [[affine space]], [[conical space]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Convex spaces have been rediscovered many times under many different names. References tend to define $c_p$ only for $0 \lt p \lt 1$, but it seems obvious that it's best to include the edge cases as well. Classically, it makes no difference, but the definition above is probably better in [[constructive mathematics]].

\begin{itemize}%
\item \emph{[[Handbook of Analysis and its Foundations]]}, Section 12.7 (short and to the point).


\item Borger \& Kemper, Cogenerators for convex spaces, Applied Categorical Struc- tures, Vol. 2 (1994), 1-11.


\item Romanowska, Smith, Orowska; Abstract barycentric algebras; \href{http://staff.science.uva.nl/~gfontain/tacl09-abstracts/tacl2009_submission_48.pdf}{pdf}. This generalises from $[0,1]$ to an arbitrary $L \Pi$-algebra ($L$ for `ukasiewicz', $\Pi$ for `product', so think of $[0,1]$ as a space of fuzzy truth values).


\item Romanowska \& Smith (1985); Modal Theory: An Algebraic Approach to Order, Geometry, and Convexity; Res. Exp. Math. 9; Heldermann-Verlag, Berlin, 1985.


\item Marshall Harvey Stone, \emph{Postulates for the barycentric calculus}, Ann. Mat. Pura. Appl. (4), 29:25--30, 1949.


\item [[Tobias Fritz]], Convex spaces I: definition and examples. \href{http://arxiv.org/abs/0903.5522}{arXiv/0903.5522}


\item [[John Baez]], [[Tobias Fritz]], [[Tom Leinster]], [[johnbaez:Convex spaces and an operadic approach to entropy]], $n$Lab draft


\item [[Bart Jacobs]], \emph{Duality for convexity} \href{http://arxiv.org/abs/0911.3834}{arXiv/0911.3834}


\item J.R. Isbell, Adequate subcategories, Illinois Journal of Math, 4, 541-552 (1960).


\item Shiri Artstein-Avidan, Vitali Milman, \emph{The concept of duality in convex analysis, and the characterization of the Legendre transform}, Annals of Math. \textbf{169}, n.2, 661-674 (2009)


\item Joe Flood, \emph{Semiconvex geometry}, J. Austral. Math. Soc. Ser. A \textbf{30} (1980/81), 496---510.


\item T. Swirszcz, \emph{Monadic functors and categories of convex sets} , Preprint No. \textbf{70}, \emph{Proc. Inst. Math. Pol. Acad. Sci.}, Warsaw; \emph{Monadic functors and convexity}, \emph{Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys.} \textbf{22} (1974), 39--42.


\item Stanley P. Gudder, \emph{Convexity and mixtures}, SIAM Review \textbf{19} (1977), 221--240; \emph{A general theory of convexity}, Milan Journal of Mathematics, \textbf{49} (1979), 89--96.


\item Xiao-qing Meng, \emph{Categories of convex sets and of metric spaces with applications to stochastic programming and related areas}, PhD thesis ([[Meng.djvu|djvu:file]])



\end{itemize}
Many other references, and a discussion of how convex spaces have been repeatedly rediscovered, can be found at the $n$-Category Caf\'e{} post \href{http://golem.ph.utexas.edu/category/2009/04/convex_spaces.html}{Convex Spaces}.

[[!redirects convex space]] [[!redirects convex spaces]] [[!redirects barycentric algebra]] [[!redirects barycentric algebras]]

[[!redirects convex combination]] [[!redirects convex combinations]] [[!redirects convex linear combination]] [[!redirects convex linear combinations]] [[!redirects convex-linear combination]] [[!redirects convex-linear combinations]]

[[!redirects convex linear function]] [[!redirects convex linear functions]] [[!redirects convex-linear function]] [[!redirects convex-linear functions]] [[!redirects convex linear map]] [[!redirects convex linear maps]] [[!redirects convex-linear map]] [[!redirects convex-linear maps]]



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