algebraic notions:
vector spaces
affine spaces
Over the reals, there are two additional notions:
Cones are the vector notion;
convex is the affine notion.
cone = closed under semipositive -sum-
convex = closed under semipositive -average-
In particular, a cone is convex.
Conversely,
Recall that you can view affine space as a hyperplane in a vector space;
given a subset of an affine space, the cone (semipositive multiples) over it
is a cone iff the subset is convex.
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Picture of:
vector n-space: R^n
affine n-space (n+1 affine dim'l!): fiber over 1 of augmentation map R^{n+1} -> R
cone: (R^{\ge 0})^n
convex: standard simplex
(intersection of affine and cone)
...do it also for R^\infty, as that's the universal example
square of theories,
and of the spaces / models
vs have *most* structure (can take sums)
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polyhedra as images of simplex!
cool, eh?
(# of dim = # of vertices)
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cone/affine/convex
You can do -exactly- the same thing with cones, affine spaces, and
convex spaces.
There the model spaces are:
- cone: $(\bR^+)^n$ (the quadrant/octant/etc.)
- affine: $\ker \bR^n \to \bR$ under sum (augmentation ideal)
- convex: intersection of these (standard simplex)
...note the containments, which correspond to containments of theories
(d'apres universal algebra, as discussed with Michael Shulman):
every vector space is a cone, etc.
...and then you can look at the various dimensions -here-.
Since they do -not- all agree, there actually is something to study!
(It's very confusing when you're given 2 different notions
but no example that separates them!)
EGs:
- an n-dimensional vector space has cone in-dimension n
and cone out-dimension n+1 (you can fit an n-dimensional model cone in
it, and can cover it with an n+1 dimensional cone)
Pf: take the usual containment to get in-dimension;
(i.e., take a vector space basis)
tack on -(v_1+...+v_n) to get out-dimension.
[careful of what you mean by in-dimension:
you can't just say "no -positive- linear relation",
as then you can fit Hilbert cone space into the line!
Instead, you need "no redundance" (remember, it's mono),
and any linear dependence yields 2 positive sides:
u+v-w = 0 -> u+v = w]
- affine spaces have exactly 1 more dimension:
this corresponds to: n+1 points in general linear position
in $\bR^n$ are (linearly) independent
- for convex sets, in-dimension = manifold dimension,
while out-dimension = number of vertices.
Isn't it cool that you can think of a hexagon
as a projection of a 5-dimensional simplex?
(or more simply a square as a projection of a tetrahedron?)