[Merged by Bors] - feature(Analysis/Convex/Basic): Convexity and the algebra map

[mans0954]

Let R and A be ordered rings, A an R-algebra and R, A, M a scalar tower. We provide sufficient conditions for an R-convex subset of M to be A-convex and for an A-convex subset of M to be R-convex.

These are used to show that, when K is RCLike and ℝ K E is a scalar tower, a set is K-convex if and only if it is ℝ-convex.

Used in #29258

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[github-actions]

PR summary 6549fa3760

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ convex_RCLike_iff_convex_real
+ convex_of_nonneg_surjective_algebraMap

You can run this locally as follows

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

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/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).

[mans0954]

The proof is inspired by Convex.convex_isRCLikeNormedField - but I couldn't find a way to deduce Convex.convex_isRCLikeNormedField from convex_of_nonneg_algebraMap.

[mans0954]

c.f. #29075

[mans0954]

CC: @Timeroot - any thoughts on this PR please?

[j-loreaux]

Certainly this seems useful, but I'm a bit unclear about the supposed extra generality.

I would like to switch some statements about Hahn-Banach to use LocallyConvexSpace 𝕜 E instead of LocallyConvexSpace ℝ E with this.

[mans0954]

Certainly this seems useful, but I'm a bit unclear about the supposed extra generality.

@j-loreaux it's an artifact of me starting this PR before I was aware of #29075 - I've removed the unnecessary extra generality now.

I would like to switch some statements about Hahn-Banach to use LocallyConvexSpace 𝕜 E instead of LocallyConvexSpace ℝ E with this.

You might also be interested in #29342 which sorts out an awkward work-around in the definition of AbsConvex.

[j-loreaux]

bors merge

[j-loreaux]

The name of this declaration is a little misleading as Function.Surjective does not appear, but I think it's a niche enough result that I'm not going to quibble further.

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• CI Success