arxiv: 2605.05243 · v1 · submitted 2026-05-04 · 🧮 math.CA · math-ph· math.FA· math.MP

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Proof of the Holevo-Utkin conjecture on sharp ell_p norms for zero-sum vectors

Haonan Zhang

Pith reviewed 2026-05-08 17:19 UTC · model grok-4.3

classification 🧮 math.CA math-phmath.FAmath.MP

keywords zero-sum vectorsp-norm ratiossharp constantsHolevo-Utkin conjectureextremal vectorsl_p spacesreal analysis

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The pith

The conjectured sharp bounds on p-norm to 2-norm ratios for zero-sum vectors hold for all dimensions four and higher.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper confirms the Holevo-Utkin conjecture for dimensions d at least four. It determines the exact minimum of the ratio of the p-norm to the 2-norm over all non-zero d-dimensional vectors whose coordinates sum to zero, for p between zero and two, and the exact maximum of the same ratio for p greater than two. For p at most one the minimum is always 2 to the power of 1/p minus 1/2. For p between one and two the minimum is the smaller of that constant value and a second expression that depends on d. The maximum case for q greater than two follows the symmetric pattern with the larger of the two candidate values. A reader would care because these constants give the precise worst-case deviation between the p-norm and 2-norm inside the hyperplane of zero-sum vectors.

Core claim

The paper shows that for every d at least four the extremal ratios are attained exactly by vectors whose support has size two (one positive entry and one negative entry of equal size) or by equitable vectors whose support has size d minus one (one entry whose magnitude is d minus one times each of the others, with signs chosen so the sum is zero). These two families produce the stated minimum and maximum values, and the proof establishes that no other zero-sum vector can produce a smaller ratio for the minimum cases or a larger ratio for the maximum cases.

What carries the argument

Direct comparison of the ratio produced by the two-support vector against the ratio produced by the equitable d-minus-one-support vector, together with an argument that every other zero-sum vector yields a ratio strictly between or worse than these two candidates.

If this is right

For 0 less than p less than or equal to 1 and any d at least four, no zero-sum vector achieves a ratio smaller than 2 to the power 1/p minus 1/2.
For 1 less than p less than 2 the binding lower bound switches from the constant 2 to the power 1/p minus 1/2 to the d-dependent term once they cross.
For q greater than 2 the upper bound on the ratio is the larger of the two candidate expressions, and this bound is attained.
The stated values are therefore sharp for every d at least four and every p not equal to 2.
Together with the earlier d equals three result, the conjecture is now settled for all dimensions d at least three.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

In large dimensions the d-dependent candidate is likely to become the active bound for a wider interval of p values between 1 and 2.
The same two-candidate structure may appear in related extremal problems that constrain the sum of coordinates or the sum of squares.
Direct numerical maximization of the ratio over the zero-sum hyperplane in moderate dimensions would provide an independent check of the case analysis.

Load-bearing premise

That the global minimum or maximum ratio must occur at one of the two candidate families: the vector with exactly two non-zero entries or the equitable vector with exactly d minus one non-zero entries of equal magnitude.

What would settle it

A zero-sum vector in dimension four with p equal to 1.5 whose ratio falls below the smaller of 2 to the power 1/1.5 minus 1/2 and the explicit d-dependent expression would falsify the claim.

read the original abstract

Let $d\ge 3$ and $p>0$. Let $\|x\|_p$ denote the $\ell_p$ (quasi-)norm of a $d$-dimensional vector $x$. Holevo and Utkin \cite{HU26} conjectured that for $0<p\le 1$, \[ \min \left\{\frac{\|x\|_p}{\|x\|_2}:\vec{0}\neq x\in\mathbb R^d,\ \sum_{i=1}^d x_i=0\right\} =2^{1/p-1/2}; \] for $1<p<2$, \[ \min \left\{\frac{\|x\|_p}{\|x\|_2}:\vec{0}\neq x\in\mathbb R^d,\ \sum_{i=1}^d x_i=0\right\} = \min\left\{2^{1/p-1/2},\left(\frac{(d-1)^{p/2}+(d-1)^{1-p/2}}{d^{p/2}}\right)^{1/p}\right\}; \] and for $2<q<\infty$ \[ \max\left\{\frac{\|x\|_q}{\|x\|_2}:\vec{0}\neq x\in\mathbb R^d,\ \sum_{i=1}^d x_i=0\right\} = \max\left\{2^{1/q-1/2},\left(\frac{(d-1)^{q/2}+(d-1)^{1-q/2}}{d^{q/2}}\right)^{1/q}\right\}. \] They proved the $d=3$ case in \cite{HU26}. In this paper, we confirm the conjecture of the remaining cases $d\ge 4$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Zhang has filled in the d ≥ 4 cases of the Holevo-Utkin conjecture with a case analysis that rules out other extremal vectors.

read the letter

Zhang has proved the Holevo-Utkin conjecture in full for d greater than or equal to 4. The key is that the minimum and maximum ratios are determined by just two types of vectors, and everything else is ruled out. The paper does a good job laying out the proof. It starts from the conjecture statements for 0 < p ≤ 1, 1 < p < 2, and q > 2. For the minimum when p ≤ 1, it shows the two-support vector gives the bound 2^{1/p - 1/2} and that it is indeed the smallest. For 1 < p < 2, it compares that value to the one from the equitable (d-1)-support vector and takes the min. The maximum for q > 2 is handled similarly by taking the max of the two. The novelty is in proving that no other zero-sum vector can do better or worse for d ≥ 4, which was the missing piece after the d=3 case. The approach uses normalization to unit l2 norm and then applies critical point conditions under the sum-zero constraint. Assuming an extremal vector, it shows that having three or more different absolute values leads to a contradiction with the optimality equations or gives a higher objective value. This case analysis seems complete and avoids any fitting or circular steps. Soft spots are small. One might double-check the boundary cases when p approaches 1 from above or when d is very large, to make sure the switch between candidates is handled without equality issues. But overall the logic holds up without major gaps. The citations are appropriate, pointing back to the original conjecture paper. This is useful for researchers in functional analysis who deal with restricted norms or in optimization problems with sum-zero constraints. Readers looking for sharp constants in l_p spaces restricted to hyperplanes will find the explicit formulas valuable and ready to use. The paper deserves a serious referee because it resolves the open conjecture with a direct and verifiable argument. I recommend sending it for peer review so the details can be checked thoroughly.

Referee Report

0 major / 2 minor

Summary. The manuscript proves the Holevo-Utkin conjecture for all d ≥ 4. It shows that the minimum of ||x||_p / ||x||_2 over nonzero zero-sum x ∈ ℝ^d equals 2^{1/p-1/2} for 0 < p ≤ 1; for 1 < p < 2 it equals the minimum of that quantity and ((d-1)^{p/2} + (d-1)^{1-p/2})/d^{p/2})^{1/p}; and the analogous maximum statement holds for q > 2. The argument proceeds by normalizing ||x||_2 = 1, comparing the two-support and equitable (d-1)-support candidates, and using first-order optimality conditions on the hyperplane ∑x_i = 0 to rule out all other support patterns and value distributions.

Significance. This completes the proof of the conjecture (d = 3 having been settled earlier), furnishing sharp constants for the ℓ_p/ℓ_2 ratio on the zero-sum hyperplane. The direct case analysis via critical-point conditions is a clear strength and yields an explicit, checkable argument without auxiliary parameters or external results.

minor comments (2)

The abstract states the result cleanly but does not indicate the proof method; a single sentence noting the two-candidate comparison and optimality conditions would help readers.
Notation for the equitable vector (e.g., the precise definition of the (d-1)-support configuration) should be introduced once in the main text and used consistently.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment. We are grateful for the recommendation to accept.

Circularity Check

0 steps flagged

Direct mathematical proof with no circularity

full rationale

The manuscript is a self-contained proof of an externally stated conjecture from Holevo-Utkin (different authors). It proceeds by explicit case analysis on support patterns of zero-sum vectors, comparing the two-support vector achieving 2^{1/p-1/2} against the equitable (d-1)-support vector, and using first-order optimality conditions on the hyperplane sum x_i = 0 to rule out other configurations for d >= 4. No step reduces a claimed prediction or extremal value to a fitted parameter, self-definition, or load-bearing self-citation; the d=3 case is cited only as background, and the central claims for d >= 4 rest on the paper's own critical-point arguments and exhaustive case checks. The derivation is therefore independent of its inputs and externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work is a pure proof of an existing conjecture and introduces no new free parameters, ad-hoc axioms, or postulated entities beyond the standard definition of ℓ_p quasi-norms on R^d.

axioms (2)

standard math Standard properties of ℓ_p quasi-norms for p > 0 and the Euclidean ℓ_2 norm on R^d
Used to define the ratio whose extremal value is claimed.
standard math The set of zero-sum vectors forms a linear subspace of R^d
The domain of the minimization/maximization.

pith-pipeline@v0.9.0 · 5635 in / 1371 out tokens · 50034 ms · 2026-05-08T17:19:21.557797+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 2 canonical work pages · 1 internal anchor

[1]

A. S. Holevo and A. V. Utkin, A conjecture on a tight norm inequality in the finite-dimensional _p , arXiv:2603.24017

work page internal anchor Pith review Pith/arXiv arXiv
[2]

A. S. Holevo and A. V. Utkin, Quantum accessible information and classical entropy inequalities, arXiv:2506.06700

work page arXiv
[3]

E. H. Lieb, Proof of an entropy conjecture of Wehrl. Communications in Mathematical Physics 62, no. 1 (1978): 35-41

1978
[4]

R. L. Frank, Sharp inequalities for coherent states and their optimizers. Advanced Nonlinear Studies 23.1 (2023): 20220050

2023