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feat(LipschitzSmooth/Convex): convex iff equivalences for K-smoothness#31

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feat(LipschitzSmooth/Convex): convex iff equivalences for K-smoothness#31
FordUniver wants to merge 1 commit into
diffbase/lipschitzSmooth-equivfrom
feat/lipschitzSmooth-equiv

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@FordUniver FordUniver commented May 19, 2026

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Adds the four pairwise iff equivalences for a differentiable convex function on a Hilbert space, between LipschitzSmoothWith K f, LipschitzWith K (fderiv ℝ f), LipschitzWith K (∇ f), and CocoerciveWith K f. The substantive work lives in the prerequisite PRs; this PR is the synthesis step that composes them.


This is the synthesis cap of the LipschitzSmooth stack: each iff direction is a one-line composition of (i) the Baillon-Haddad hard direction LipschitzSmoothWith K f → CocoerciveWith K f under convexity, (ii) the non-convex Hilbert-form descent lemma LipschitzWith K (∇ f) → LipschitzSmoothWith K f, (iii) the elementary direction CocoerciveWith K f → LipschitzWith K (∇ f) (Cauchy-Schwarz), and (iv) the Riesz isomorphism LipschitzWith K (fderiv ℝ f) ↔ LipschitzWith K (∇ f). Splitting these out of the Baillon-Haddad PR keeps that PR focused on the actual theorem and isolates the dependency on the descent lemma to this synthesis layer.

Diff for the changes just in this PR over its predecessors: link

@FordUniver FordUniver force-pushed the diffbase/lipschitzSmooth-equiv branch from 3a654e2 to 9d8c923 Compare May 19, 2026 11:10
@FordUniver FordUniver force-pushed the feat/lipschitzSmooth-equiv branch from b4eb412 to 735a2fc Compare May 19, 2026 11:10
Add the four pairwise iff equivalences for a differentiable convex function on
a Hilbert space — `LipschitzSmoothWith`, `LipschitzWith (fderiv ℝ f)`,
`LipschitzWith (∇ f)`, `CocoerciveWith` — synthesising the Baillon-Haddad
hard direction (`LipschitzSmoothWith K f → CocoerciveWith K f`), the
non-convex Hilbert-form descent lemma, and the elementary direction
`CocoerciveWith → LipschitzWith (∇ f)`.
@FordUniver FordUniver force-pushed the diffbase/lipschitzSmooth-equiv branch from 9d8c923 to 33bd89d Compare May 19, 2026 11:38
@FordUniver FordUniver force-pushed the feat/lipschitzSmooth-equiv branch from 735a2fc to 1204249 Compare May 19, 2026 11:39
@FordUniver

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Folded back into #29 — the four iffs are 30 lines of pure compositions of (i) Baillon-Haddad and (ii) the descent lemma in Hilbert form, and conceptually inseparable from the BH theorem itself. Splitting them was over-application of 'minimize formal dependencies'.

@FordUniver FordUniver closed this May 19, 2026
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