The standard simplex #
In this file, given an ordered semiring 𝕜
and a finite type ι
,
we define stdSimplex : Set (ι → 𝕜)
as the set of vectors with non-negative
coordinates with total sum 1
.
The standard simplex in the space of functions ι → 𝕜
is the set of vectors with non-negative
coordinates with total sum 1
. This is the free object in the category of convex spaces.
Instances For
The standard simplex in the zero-dimensional space is empty.
All values of a function f ∈ stdSimplex 𝕜 ι
belong to [0, 1]
.
The edges are contained in the simplex.
The standard one-dimensional simplex in Fin 2 → 𝕜
is equivalent to the unit interval.
This bijection sends the zeroth vertex Pi.single 0 1
to 0
and
the first vertex Pi.single 1 1
to 1
.
Equations
Instances For
stdSimplex 𝕜 ι
is the convex hull of the canonical basis in ι → 𝕜
.
stdSimplex 𝕜 ι
is the convex hull of the points Pi.single i 1
for i :
i`.
The convex hull of a finite set is the image of the standard simplex in s → ℝ
under the linear map sending each function w
to ∑ x ∈ s, w x • x
.
Since we have no sums over finite sets, we use sum over @Finset.univ _ hs.fintype
.
The map is defined in terms of operations on (s → ℝ) →ₗ[ℝ] ℝ
so that later we will not need
to prove that this map is linear.
Every vector in stdSimplex 𝕜 ι
has max
-norm at most 1
.
stdSimplex ℝ ι
is bounded.
stdSimplex ℝ ι
is closed.
stdSimplex ℝ ι
is compact.
The standard one-dimensional simplex in ℝ² = Fin 2 → ℝ
is homeomorphic to the unit interval.
Equations
- One or more equations did not get rendered due to their size.