Affine Structures  

Affine Space > s.a. vector space.
$ Def: An affine space of dimension n over \(\mathbb R\) (or a vector space V) is a set E on which the additive group \(\mathbb R\)n (or V) acts simply transitively.
* Idea: It can be considered as a vector space without an origin (therefore without preferred coordinates, addition and multiplication by a scalar); If v is an element of \(\mathbb R\)n (or V), we can write things like P = Q + v, or v = PQ, but we cannot give a meaning to P + Q.
* Examples: Any vector space is an affine space over itself, with composition being vector addition.
* Affine subspace: A subset SE of the form S = p + X, with pE and X a vector subspace of V; It is an affine space over X.
* Compatible topology: A topology on E compatible with translations, in the sense that all maps p \(\mapsto\) p + v are continuous; Any such topology on E can be induced (as a "translated topology") by one on V; If V is a finite-dimensional real or complex vector space, there is a unique, natural topology on it, and therefore on E as well, which is first countable.
@ General references: in Greenberg & Harper 81; Kostrikin & Manin 89.
> Online resources: see Wikipedia page.

Convex Subset of an Affine Space
$ Def: A subset A of an affine space X is said to be convex iff the segment connecting x and y is contained in A,

x, yA,      λ x + (1–λ) yA, ∀ λ ∈ [0,1] .

* Support function: For a convex subset K of Euclidean space bounded by γ, the support function p(φ) gives the distance p to the origin of the line tangent to the curve γ with angle φ with respect to the x axis.
* Area of convex 2D set: For a convex K ⊂ \(\mathbb R\)2,

A(K) = \(1\over2\)\(\displaystyle\oint_{\partial K}\)p(φ) ds = \(1\over2\)\(\displaystyle\oint_0^{2\pi}\)(p2p' 2) dφ .

* Star-convex subset: A subset AX is star-convex relative to x0X iff ∀xA, λx + (1–λ) x0A, ∀λ ∈ [0,1]; Can be seen as the union of segments with one endpoint at x0; It is typically star-shaped around x0.
* Properties: Any convex set is star-convex relative to any of its points; The set of convex subsets is stable under intersections, but not under unions.
* Relationships: For any reasonable topology (e.g., Euclidean), all convex K are sequentially connected, and all sequentially connected K are connected.

Affine Map / Transformation
* Idea: A map that preserves the affine structure on a space.
$ Def: In \(\mathbb R\)n, a composition x' = A x + a of non-singular linear transformations and translations.
* Properties: It preserves geometrically independent points.

Affine Structure > s.a. affine connection; differential geometry [affine manifold]; projective structure; Weyl Space.
$ Def: A differentiable manifold with a preferred set of affinely parametrized geodesics – an affine geometry at each point.
* Idea: Provides a notion of parallel propagation of a vector along any curve – expressed by an affine connection.
* Relationships: It implies a projective structure.

Related Concepts
$ Geometrically independent points: The points a0, ..., an ∈ \(\mathbb R\)n are geometrically independent if, for any given set t0, ..., tn of scalars, ∑i ti = 0 and ∑i ti ai = 0 implies ti = 0, for all i; In RN: There can be at most N + 1 geometrically independent points; Using n of them as vertices, one can make a simplex; An affine transformation can take these points into the origin and the tips of the first n unit vectors.
@ Affine collineation: Hall & da Costa JMP(88); Hojman & Núñez JMP(91) [Riemannian manifolds].
> Physics topics: see affine connections; Affine Gravity; modified coherent states; quantum formalism [affine variables].

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