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*
= *P* − *Q*, 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
*S* ⊂ *E* of the form *S* = *p* + *X*,
with *p* ∈ *E* 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*, *y* ∈ *A*, *λ* *x*
+ (1−*λ*) *y* ∈ *A*, ∀ *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*)
d*s* = \(1\over2\)\(\displaystyle\oint_0^{2\pi}\)(*p*^{2}
− *p'*^{ 2}) d*φ* .

* __Star-convex subset__:
A subset *A* ⊂ *X* is star-convex relative to
*x*_{0} ∈ *X* iff ∀*x* ∈ *A*,
*λx* + (1−*λ*)
*x*_{0} ∈ *A*, ∀*λ* ∈ [0,1];
Can be seen as the union of segments with one endpoint at *x*_{0};
It is typically star-shaped around *x*_{0}.

* __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 *a*_{0}, ...,
*a*_{n} ∈ \(\mathbb R^n\) are
geometrically independent if, for any given set *t*_{0},
..., *t*_{n} of scalars,
∑_{i} *t*_{i}
= 0 and ∑_{i} *t*_{i}
*a*_{i} = 0 implies
*t*_{i} = 0, for all *i*;
__In R__

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send feedback and suggestions to bombelli at olemiss.edu – modified 7 aug 2018