Alphabeta MathConvexity: Examples and Counterexamples
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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.
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✓ 1 result · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 1 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convexity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
The absolute-value function is convex
Example
Finite Jensen for x↦ x² gives that the square of a weighted mean is at most the weighted mean of the squares
Counterexample
Assuming Choice, a Hamel coefficient map is midpoint convex but discontinuous and therefore not convex
Example