Let p≥ 2 be an even integer and let (a_1,a_2,…,a_n)∈R^n satisfy ∑_k a_k^2=1. Let X be a symmetric random variable, let X_1,X_2,…,X_n be independent identically distributed copies of X, and write r_k=r_k(X). Define S=∑_k a_kX_k,qquad Y_n=frac1√ n(X_1+X_2+…+X_n). Diagonal stability conjecture. There exist positive constants C_p>0 such that, if (r_k) is log-concave, then |S|_p^p≤ |Y_n|_p^p-C_p∑_k(a_k^2-frac1n)^2, and, if (r_k) is log-convex, then |S|_p^p≥ |Y_n|_p^p+C_p∑_k(a_k^2-frac1n)^2. This conjecture proposes a quantitative stability estimate for the diagonal case of Schur-type Khintchine inequalities: the gap from the equal-weight vector is controlled by the squared deviations of the coefficients from 1/n. The paper leaves it as an open problem.

Diagonal stability conjecture for Khintchine-type inequalities

Let $p\geq 2$ be an even integer and let $(a_1,a_2,\ldots,a_n)\in\mathbb{R}^n$ satisfy

\sum_k a_k^2=1.

Let $X$ be a symmetric random variable, let $X_1,X_2,\ldots,X_n$ be independent identically distributed copies of $X$ , and write $r_k=r_k(X)$ . Define

S=\sum_k a_kX_k,\qquad Y_n=\frac{1}{\sqrt n}(X_1+X_2+\cdots+X_n).

Diagonal stability conjecture. There exist positive constants $C_p>0$ such that, if $(r_k)$ is log-concave, then

\|S\|_p^p\leq \|Y_n\|_p^p-C_p\sum_k\left(a_k^2-\frac{1}{n}\right)^2,

and, if $(r_k)$ is log-convex, then

\|S\|_p^p\geq \|Y_n\|_p^p+C_p\sum_k\left(a_k^2-\frac{1}{n}\right)^2.

This conjecture proposes a quantitative stability estimate for the diagonal case of Schur-type Khintchine inequalities: the gap from the equal-weight vector is controlled by the squared deviations of the coefficients from $1/n$ . The paper leaves it as an open problem.

References

Primary source

Ángel Chávez and Sam Sheng, “Stability of Khintchine-type inequalities via log-monotonicity”, arXiv:2606.19313 (2026).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

The paper leaves the conjecture open, but an unverified posted calculation claims a complete Gaussian counterexample that would refute both inequalities.

Chávez and Sheng formulated the diagonal stability conjecture in their June 2026 paper, asking for a positive quadratic gap from equal coefficients for every even $p\geq 2$ . They explicitly leave the general problem open.

Known results

Rademacher variables: Jakimiuk proved the log-concave inequality with a positive quadratic deficit for every $p\geq 4$ .

Posted attempt

A posted attempt claims a complete disproof: for $X$ standard Gaussian, $n=2$ , and coefficients $(1,0)$ , both $S$ and $Y_2$ are identically distributed, while the proposed deviation term is positive. It also notes the universal $p=2$ obstruction and gives an exact $p=4$ identity. This attempt has not been independently verified.

Current status (as of August 2026): The published source records the general conjecture as open; a complete counterexample has been claimed in discussion but remains unverified.

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

Both proposed diagonal-stability inequalities are false for every even p≥2, even for a smooth, nondegenerate, unit-variance Gaussian distribution.

The source defines

r_j(X)=j! E[X^{2j}]/(2j)!.

Take X=G∼N(0,1), n=2, and (a₁,a₂)=(1,0). Since

E[G^{2j}]=(2j)!/(2^j j!),

we have

r_j(G)=2^{−j},

which is simultaneously log-concave and log-convex:

r_j(G)²=r_{j−1}(G)r_{j+1}(G).

Thus both stated hypotheses apply. But

S=G₁, Y₂=(G₁+G₂)/√2

have the same standard Gaussian distribution, while

(a₁²−1/2)²+(a₂²−1/2)²=1/2.

Writing m_p=E|G|^p, the two proposed conclusions become respectively

m_p≤m_p−C_p/2,

m_p≥m_p+C_p/2.

Both are impossible for every C_p>0 and every even p≥2. This remains false even if C_p is permitted to depend on the distribution.

There is an independent universal endpoint obstruction. For every symmetric finite-variance X and every normalized coefficient vector,

E|S|²=E[X²]=E|Y_n|².

Hence at the included exponent p=2, any nonuniform squared-weight vector contradicts either claimed positive deficit, irrespective of log-monotonicity.

At p=4 there is also a sharp corrected theorem. Put μ₂=E[X²], μ₄=E[X⁴], and

D(a)=∑_{i=1}^n(a_i²−1/n)²=∑_i a_i⁴−1/n.

Independence gives the exact identity

E[S⁴]−E[Y_n⁴]=(μ₄−3μ₂²)D(a).

Because r₀=1, r₁=μ₂/2, and r₂=μ₄/12, log-concavity implies μ₄≤3μ₂² and log-convexity implies μ₄≥3μ₂². The corresponding optimal fixed-distribution stability constants are therefore

C₄=3μ₂²−μ₄ in the log-concave case,

C₄=μ₄−3μ₂² in the log-convex case.

They are strictly positive exactly when the relevant kurtosis inequality is strict. For the Gaussian, both vanish, explaining precisely why the weak log-concavity/log-convexity assumptions in Conjecture 7.1 cannot imply a positive stability constant.