CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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.

A strictly convex function can have a singular Hessian

Statement refuted

Every twice differentiable strictly convex function has a positive-definite Hessian at every point.

The counterexample below establishes: For every $n \geq 1$ , the function $f : R^{n} \to R$ given by $f (x) = \sum_{j < n} x_{j}^{4}$ is strictly convex, but its Hessian at the origin is the zero matrix.

Facts & Assumptions

Given: Fix $n \geq 1$ . Strict convexity is as in Convex and strictly convex functions on Euclidean convex sets, and the Hessian convention is The Hessian matrix and critical points of a scalar field.

[L1]

For every integer $d \geq 1$ , the function $p_{d} (x) = x^{d}$ is differentiable everywhere and satisfies $p_{d}^{'} (x) = d x^{d - 1}$ (For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term).

[L2]

For a continuous real function on a closed interval that is differentiable in its interior, one secant slope equals an interior derivative (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

[L3]

Scalar multiples of differentiable real functions are differentiable with the expected derivatives (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

Counterexample

technique · direct

1.1

Let $r < s$ and $0 < t < 1$ , and put $c = (1 - t) r + t s$ . Apply [L2] on $[r, c]$ and $[c, s]$ . The two secant slopes equal $4 u^{3}$ and $4 v^{3}$ for some $r < u < c < v < s$ by [L1], so the first is strictly smaller; rearranging gives $c^{4} < (1 - t) r^{4} + t s^{4}$ . Thus $x^{4}$ is strictly convex. For distinct vectors, at least one coordinate gives this strict inequality and every other coordinate gives the corresponding weak one; adding shows that $f (x) = \sum_{j < n} x_{j}^{4}$ is strictly convex for $n \geq 1$ .

L1L2givenalgebra

2.1

Differentiating again with [L1] and [L3], the Hessian is diagonal with entries $12 x_{j}^{2}$ and zero off-diagonal entries. At the origin it is the zero matrix, hence singular and not positive definite because $n \geq 1$ . Thus $f (x) = \sum_{j < n} x_{j}^{4}$ is strictly convex, but its Hessian at the origin is the zero matrix.

L1L3step 1.1givenalgebra∎

Depends on

Used by

FALSE: strict convexity gives a positive-definite Hessian False statement

Dependency tree · two levels

26 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, §3.1.4 (standard reference, not scraped)