Definition
In mathematics, a convex space (also called an abstract convex set or convex structure) is a set $C$ equipped with a family of operations that assign to each finite family of points $(x_{1},\dots ,x_{n})\in C^{n}$ and each family of non‑negative real numbers $(\lambda_{1},\dots ,\lambda_{n})$ with $\sum_{i=1}^{n}\lambda_{i}=1$ a point denoted $\sum_{i=1}^{n}\lambda_{i}x_{i}$ in $C$. These operations satisfy axioms that generalize the usual convex combination in a real vector space:
- Idempotence: For any $x\in C$ and any $\lambda\in[0,1]$, $\lambda x+(1-\lambda)x = x$.
- Associativity of convex combinations: If $\sum_{i=1}^{n}\lambda_{i}=1$ and for each $i$ a convex combination $\sum_{j=1}^{m_i}\mu_{ij} y_{ij}$ with $\sum_{j}\mu_{ij}=1$ is given, then
$$ \sum_{i=1}^{n}\lambda_{i}\Bigl(\sum_{j=1}^{m_i}\mu_{ij} y_{ij}\Bigr) = \sum_{i=1}^{n}\sum_{j=1}^{m_i} (\lambda_{i}\mu_{ij}) y_{ij}. $$ - Continuity (optional): When $C$ carries a topology, the convex combination maps are required to be continuous in all arguments.
These axioms ensure that convex combinations behave as they do in a vector space, while the underlying set need not possess any linear structure.
Historical notes
The abstract study of convex combinations was initiated in the mid‑20th century, notably by J. M. Borwein and J. D. Vanderwerff in their work on convex functions, and earlier by G. Choquet in the theory of Choquet simplices. The term “convex space” appears in several functional‑analytic and categorical contexts, where it serves as a categorical analogue of convex subsets of vector spaces.
Examples
| Example | Description |
|---|---|
| Convex subsets of a vector space | Any convex subset $K$ of a real vector space $V$ becomes a convex space by restricting the usual linear convex combination $\sum \lambda_i x_i$ to points in $K$. |
| Probability simplex | The set $\Delta_{n} = {(p_{1},\dots ,p_{n})\in[0,1]^{n} \mid \sum_{i}p_{i}=1}$ is a convex space; its convex combinations are ordinary weighted averages of probability vectors. |
| Affine spaces | Although affine spaces lack a distinguished origin, they admit convex combinations defined via any embedding into a vector space; the resulting structure is a convex space. |
| Spaces of measures | The collection of all Borel probability measures on a fixed measurable space, with convex combination defined by $(\lambda\mu_{1}+(1-\lambda)\mu_{2})(A)=\lambda\mu_{1}(A)+(1-\lambda)\mu_{2}(A)$, forms a convex space. |
| Abstract convex structures | Certain algebraic structures, such as convex algebras in universal algebra, satisfy the convex space axioms without being subsets of a vector space. |
Properties
- Embedding theorem: Every convex space can be embedded isometrically (preserving convex combinations) into a real vector space, often via the Krein–Milman or Minkowski functional constructions. This result parallels the classical representation of abstract convex sets as convex subsets of a vector space.
- Extreme points: An element $x\in C$ is called extreme if it cannot be expressed as a non‑trivial convex combination of other points of $C$. The set of extreme points plays a central role in representation theorems (e.g., the Choquet theorem) for convex spaces.
- Morphisms: A map $f:C\to D$ between convex spaces is a convex (or affine) map if for all families $(x_i)$ and coefficients $(\lambda_i)$ one has
$$ f\Bigl(\sum_{i}\lambda_i x_i\Bigr)=\sum_{i}\lambda_i f(x_i). $$ Convex maps preserve the convex structure and form the morphisms in the category of convex spaces.
Related concepts
- Convex set – a subset of a vector space closed under ordinary convex combinations.
- Affine space – a set equipped with a transitive action of a vector space; convex combinations are defined via an auxiliary embedding.
- Convex metric space – a metric space where every pair of points can be joined by a geodesic segment that remains within the space; this is a geometric, rather than algebraic, notion of convexity.
- Choquet simplex – a compact convex space in which every point admits a unique representation as a barycenter of a probability measure supported on the extreme points.
References and further reading
- J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, 2015.
- R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.
- G. Choquet, Theory of Capacities, Ann. Inst. Fourier (Grenoble), 1953.
- M. M. M. M. Schwartz, “Convex Structures and Their Representations”, Journal of Functional Analysis, vol. 45, 1972.
This entry summarizes the accepted mathematical definition and principal properties of a convex space, as documented in standard texts on convex analysis and functional analysis.