Alphabeta MathHow 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.
FALSE: every convex function is differentiable
Statement
False claim: every convex real-valued function on an open Euclidean convex set is differentiable everywhere.
Facts & Assumptions
Given: Positive Euclidean dimension.
The Euclidean norm is convex and its subdifferential at zero is the closed unit ball (The Euclidean norm and its square are convex, with a ball of subgradients at zero for the norm).
If a convex function is differentiable at , then (The subdifferential of a differentiable convex function is its gradient singleton).
Refutation
In positive dimension, the closed unit ball in [L1] contains more than one vector, so the norm has a nonsingleton subdifferential at zero.
By [L2], differentiability there would force the subdifferential to be a singleton. Hence the convex Euclidean norm is not differentiable at zero, and the claim is false.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
13 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
- S. Boyd and L. Vandenberghe, Convex Optimization, §3.1 (standard reference, not scraped)
- D. Bertsekas, MIT 6.253 Convex Analysis and Optimization, Lecture 12 (standard reference, not scraped)