TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Alexandrov second differentiability of locally semiconvex functions

Statement

Assume the Axiom of Choice. Let $U \subseteq R^{n}$ be open, $n \geq 1$ , and let $u : U \to R$ be locally semiconvex: around each point there is a Euclidean ball (Open ball, closed ball and sphere in a metric space) and $c \geq 0$ such that $x \mapsto u (x) + c ∣ x ∣^{2} / 2$ is convex in the sense of Convex and strictly convex functions on Euclidean convex sets. Outside a set null in the cube-cover sense of Measure zero and content zero in $R^{m}$ by countable and finite cube covers, there are $p \in R^{n}$ and a symmetric matrix $A$ with $u (x + h) = u (x) + p \cdot h + \frac{1}{2} h^{T} A h + o (∣ h ∣^{2}) (h \to 0) .$ The remainder is uniform in direction. In particular $u$ is twice differentiable in the Peano sense almost everywhere.

Facts & Assumptions

Given: The Axiom of Choice, an open set $U \subseteq R^{n}$ , $n \geq 1$ , and a locally semiconvex real function $u$ on $U$ .

[A1]

Every family of nonempty sets has a choice function (The Axiom of Choice); the implication AC $\Rightarrow$ DC $\Rightarrow$ Countable Choice is supplied by AC implies DC implies countable choice, discharging both choice hypotheses of the absolutely-continuous FTC in [F7].

[F1]

A locally Lipschitz map between finite-dimensional Euclidean spaces is Frechet differentiable outside a cube-cover-null set (Rademacher differentiability in Euclidean space).

[F2]

A Lipschitz map $R^{n} \to R^{n}$ sends null sets to null sets (A Lipschitz map $R^{m} \to R^{m}$ sends null sets to null sets).

[F3]

The image of the differentiability points with singular derivative under a Lipschitz self-map of $R^{n}$ is null (Critical images of Euclidean Lipschitz maps are null).

[F4]

For a finite convex function on an open convex set, the subdifferential is nonempty, locally bounded, has locally closed graph, and its singleton points are exactly its differentiability points; the localized convex extension and dual-gradient inverse also hold (Convex supports, localization, and the Lipschitz dual gradient).

[F5]

Every bounded sequence in $R^{n}$ has a convergent subsequence (For $n \geq 1$ every bounded sequence in $R^{n}$ has a convergent subsequence).

[F6]

Countable Choice is the choice function principle for countable families (The Axiom of Countable Choice ( ${A C}_{ω}$ )); under it, countable unions of null subsets of $R^{n}$ are null (Subsets and countable unions of null subsets of $R^{m}$ are null).

[F7]

Lipschitz functions on intervals are absolutely continuous, and absolutely continuous functions are integrals of their almost-everywhere derivatives ( $C^{1}$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation, Fundamental theorem of calculus for absolutely continuous functions).

[F8]

Nullity in $R^{n}$ is defined by arbitrarily small total volumes of closed-cube covers (Measure zero and content zero in $R^{m}$ by countable and finite cube covers).

Proof

1.1A1F1F2F3F4F6F8construct

First consider a finite convex function $f$ on an open ball. Localize on a smaller closed ball $K$ and use [F4] to obtain a finite globally Lipschitz convex $H = f$ on $K$ , then set $F = H + ∣ \cdot ∣^{2} / 2$ and $g = D F^{*}$ , which is globally one-Lipschitz and satisfies $g (q) = x$ exactly when $q \in \partial F (x)$ . By [F1], the nondifferentiability set $N$ of $g$ is null; [F2] makes $g (N)$ null, and [F3] makes $g (C)$ null for the differentiable singular-derivative set $C$ of $g$ . The locally Lipschitz function $F$ is differentiable outside another null set $M$ by [F1]. The union $M \cup g (N) \cup g (C)$ is null by [F6]. For $x$ in the interior of $K$ outside that union, set $p = D F (x)$ . Then $g (p) = x$ , $g$ is differentiable at $p$ , and $B = D g (p)$ is invertible. Full AC is used by the stated suppliers and to choose the localizations simultaneously.

2.1A1F4F5step 1.1algebra

As $y \to x$ , every $q \in \partial F (y)$ tends to $p$ , uniformly over all such subgradients: otherwise, by [A1], choose $y_{k} \to x$ and $q_{k} \in \partial F (y_{k})$ staying a fixed distance from $p$ ; local boundedness in [F4] and [F5] give a convergent subsequence, and the locally closed graph in [F4] puts its limit in $\partial F (x) = {p}$ , a contradiction. Since $g (q) = y$ , differentiability of $g$ at $p$ gives $y - x = B (q - p) + o (∣ q - p ∣)$ . Invertibility absorbs the remainder to give $∣ q - p ∣ \leq 2 ∥ B^{- 1} ∥ ∣ y - x ∣$ for nearby $y$ , hence $q = p + B^{- 1} (y - x) + o (∣ y - x ∣)$ uniformly for every $q \in \partial F (y)$ .

3.1F7step 2.1algebra

Fix a small $h$ . The section $t \mapsto F (x + t h)$ is Lipschitz on $[0, 1]$ , hence absolutely continuous by [F7]. Wherever its derivative exists, the two support inequalities in directions $h$ and $- h$ show that it equals $q \cdot h$ for every $q \in \partial F (x + t h)$ . By step 2.1, for every $ϵ > 0$ and sufficiently small $∣ h ∣$ , $∣ q - p - B^{- 1} (t h) ∣ \leq ϵ t ∣ h ∣$ for every $0 < t \leq 1$ and every such $q$ . Integrating the measurable scalar derivative using [F7] gives $∣ F (x + h) - F (x) - p \cdot h - \frac{1}{2} h^{T} B^{- 1} h ∣ \leq ϵ ∣ h ∣^{2} / 2.$ No measurable subgradient selection is used. Replace $B^{- 1}$ by its symmetric part, which has the same quadratic form.

4.1A1F6F8step 3.1construct∎

Subtract $∣ x ∣^{2} / 2$ to obtain the quadratic expansion for $f$ in the interior of $K$ . Exhaust the original ball by countably many smaller closed balls and discard their null exceptional sets using [F6, F8]. The family of rational balls whose closures lie inside a local semiconvexity ball is countable and covers $U$ ; by [A1] choose a corresponding finite convexity constant for each. Apply the convex conclusion to $u + c ∣ \cdot ∣^{2} / 2$ on each ball and subtract that quadratic. Countable unions remain null by [F6, F8], and the remainder is direction-uniform by step 3.1. Full AC is retained; no reduction to Countable Choice is asserted.

Depends on

Used by

Elliptic theorem of sums with the full block matrix bound Theorem

Dependency tree · two levels

92 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Juha Kinnunen, Sobolev Spaces, complete 168-page notes; Sections 2.6 and 3.3-3.4 (standard reference, not scraped)
Crandall, Ishii and Lions, User's guide to viscosity solutions, Appendix A, Theorem A.2 and pp. 58-60, notes pp. 62-63 (standard reference, not scraped)