feat(Geometry/Convex/ConvexSpace): show that an AffineMap IsAffineMap

[ooovi]

Show that Convexity.IsAffineMap generalises AffineMaps between affine spaces.

---

[github-actions]

PR summary 46c86287c1

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

+ isAffineMap
+ of_apply_affineCombination

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 46c86287c1
Reference commit c87658adb9

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[ooovi]

I hope did the correct thing with the local instance...?

[YaelDillies]

Suggested change

	lemma of_apply_affineCombination
	lemma of_map_affineCombination

is the naming convention when the applied function is a free variable (or the coercion to function of a free variable)

See leanprover-community/leanprover-community.github.io#886

[YaelDillies]

Suggested change

	/-- A map between convex spaces is affine if it preserves convex combinations. -/
	/-- A map between convex spaces is affine if it preserves convex combinations.
	Note that this generalises the notion of affine maps between affine spaces.
	See `AffineMap.isAffineMap` for one direction.
	TODO: Prove the other direction. -/

[YaelDillies]

Suggested change

	IsAffineMap k f where
	IsAffineMap k f where

[Paul-Lez]

One small note: I wasn't very satisfied with the universe for ι in this line, which is why my first suggestion was

lemma AffineMap.isAffineMap {V2 P2 : Type*} [AddCommGroup V2] [Module R V2] [AffineSpace V2 P2]
    (f : P →ᵃ[R] P2) : IsAffineMap R f  where
  map_sConvexComb s := by
     rw [sConvexComb_eq_affineCombination, map_affineCombination s.weights.support _root_.id
      s.weights s.total, ←iConvexComb_eq_affineCombination, Function.comp_id, iConvexComb]

Perhaps here the correct thing to do would be to just use Type?