Convex sets and functions in vector spaces #
In a 𝕜-vector space, we define the following objects and properties.
Convex 𝕜 s: A set
sis convex if for any two points
x y ∈ sit includes
segment 𝕜 x y.
stdSimplex 𝕜 ι: The standard simplex in
ι → 𝕜(currently requires
Fintype ι). It is the intersection of the positive quadrant with the hyperplane
s.sum = 1.
We also provide various equivalent versions of the definitions above, prove that some specific sets are convex.
Generalize all this file to affine spaces.
Convexity of sets #
Alias of the forward direction of
Alternative definition of set convexity, in terms of pointwise set operations.
Alternative definition of set convexity, using division.