Definition
In geometry and linear algebra, the affine hull of a set $S$ of points in a real vector space $V$ is the smallest affine subspace of $V$ that contains $S$. Equivalently, it is the set of all affine combinations of points in $S$. An affine combination of points $x_1, x_2, \dots, x_k \in S$ is a linear combination
$$ \sum_{i=1}^{k} \lambda_i x_i \quad\text{with}\quad \sum_{i=1}^{k} \lambda_i = 1, $$
where $\lambda_i \in \mathbb{R}$.
Formally, $$ \operatorname{aff}(S)=\left{\sum_{i=1}^{k}\lambda_i x_i ;\bigg|; x_i\in S,; \lambda_i\in\mathbb{R},; \sum_{i=1}^{k}\lambda_i=1,; k\in\mathbb{N}\right}. $$
Key Properties
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Minimal Affine Set – $\operatorname{aff}(S)$ is an affine subspace containing $S$; any affine subspace containing $S$ must also contain $\operatorname{aff}(S)$.
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Relation to Linear Span – If $a\in S$ and we translate the set by $-a$, then
$$ \operatorname{aff}(S)=a+\operatorname{span}(S-a), $$ where $\operatorname{span}(S-a)$ denotes the linear span of the translated set ${x-a \mid x\in S}$. -
Dimension – The dimension of the affine hull equals the dimension of the linear span of the differences of points in $S$. For a finite set of $k$ affinely independent points, $\dim\operatorname{aff}(S)=k-1$.
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Closedness – In finite‑dimensional Euclidean spaces, $\operatorname{aff}(S)$ is a closed set. In infinite‑dimensional spaces, closedness depends on the topology of the underlying space.
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Convex Hull Inclusion – The convex hull $\operatorname{conv}(S)$ is always a subset of the affine hull:
$$ \operatorname{conv}(S) \subseteq \operatorname{aff}(S). $$
Equality holds when $S$ is already convex and affinely closed. -
Preservation under Affine Maps – If $f:V\to W$ is an affine map (i.e., $f(x)=Ax+b$ with linear $A$ and vector $b$), then
$$ f\bigl(\operatorname{aff}(S)\bigr)=\operatorname{aff}\bigl(f(S)\bigr). $$
Examples
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Two points $x, y$ in $\mathbb{R}^n$:
$\operatorname{aff}{x,y}$ is the straight line passing through $x$ and $y$. -
Three non‑collinear points in $\mathbb{R}^3$:
Their affine hull is the unique plane containing the three points. -
A single point $x$:
$\operatorname{aff}{x}= {x}$, a zero‑dimensional affine subspace. -
All of $\mathbb{R}^n$:
If $S$ contains $n+1$ affinely independent points, then $\operatorname{aff}(S)=\mathbb{R}^n$.
Applications
- Computational Geometry – Determining the affine hull is a preprocessing step for algorithms involving convex hulls, Voronoi diagrams, and linear programming.
- Optimization – In linear programming, feasible regions are often described as intersections of affine hulls with half‑spaces.
- Data Analysis – Principal Component Analysis (PCA) can be interpreted as finding the low‑dimensional affine subspace that best approximates a data set in a least‑squares sense.
Related Concepts
- Affine subspace: a translation of a linear subspace; the affine hull is the smallest such subspace containing a given set.
- Convex hull: the smallest convex set containing the original set; always contained within the affine hull.
- Linear span: the set of all linear combinations of a set; the affine hull is obtained by allowing coefficients that sum to one, effectively adding a translation component.
References
Standard textbooks on linear algebra and convex geometry, such as:
- R. Rockafellar, Convex Analysis, Princeton University Press, 1970.
- D. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Cambridge University Press, 2014.
(These works provide rigorous proofs of the properties listed above.)
