Proof of the Holevo--Utkin conjecture on sharp ell_p norms for zero-sum vectors
Pith reviewed 2026-05-22 11:08 UTC · model grok-4.3
The pith
Explicit formulas give the exact minimum and maximum ratios of ℓ_p to ℓ_2 norms for non-zero zero-sum vectors when the dimension is four or higher.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For d greater than or equal to four the minimum of ||x||_p / ||x||_2 equals 2^{1/p-1/2} for 0 < p ≤ 1, equals the minimum of that quantity and the p-th root of ((d-1)^{p/2} + (d-1)^{1-p/2}) / d^{p/2} for 1 < p < 2, and the maximum for q > 2 is the maximum of the analogous two quantities.
What carries the argument
Vectors whose non-zero entries take at most two distinct values, such as one entry versus d-1 equal entries or two equal-magnitude opposite entries.
If this is right
- The given expressions are the best possible constants for every d at least 4 and every p greater than zero.
- For p between 1 and 2 the minimum switches from one expression to the other at a critical value that depends on d.
- The same simple vectors achieve the bound uniformly in all higher dimensions.
- No zero-sum vector can produce a ratio outside the stated min or max in the respective ranges of p.
Where Pith is reading between the lines
- The same candidate vectors could be checked for extremality under a different linear constraint instead of exact zero sum.
- As d grows the second expression approaches 1, which might simplify asymptotic bounds for large-dimensional balanced systems.
Load-bearing premise
The extremal ratio is attained only by vectors whose non-zero entries take at most two distinct values.
What would settle it
A zero-sum vector in R^4 for p=1.5 whose ratio ||x||_{1.5} / ||x||_2 is strictly smaller than the minimum of the two explicit expressions would disprove the claimed sharp bound.
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.
Referee Report
Summary. The manuscript proves the Holevo-Utkin conjecture for all d ≥ 4. It establishes that the extremal ratio ||x||_p / ||x||_2 (with ||x||_2 = 1 and sum x_i = 0) is attained only by the two families of vectors already identified in the abstract: the one-nonzero-entry vectors (one entry versus d-1 equal entries) and the two-support sign-alternating vectors. For 0 < p ≤ 1 the minimum is 2^{1/p-1/2}; for 1 < p < 2 the minimum is the smaller of the two candidate expressions; the corresponding maximum is given for 2 < q < ∞. The argument proceeds by direct comparison and by invoking strict convexity/concavity of t ↦ |t|^p on the relevant intervals.
Significance. The result completes the proof of the conjecture for every dimension d ≥ 3 and supplies explicit, sharp constants. The reduction to the two extremal families is carried out explicitly in the main theorem and supporting lemmas, with no hidden assumptions left unproved. This supplies a parameter-free, falsifiable characterization that can be used directly in applications.
minor comments (2)
- [Main theorem] Main theorem: the case distinctions for p (0<p≤1, 1<p<2, 2<q<∞) are handled separately; a single consolidated statement with explicit min/max would improve readability.
- [Supporting lemmas] Supporting lemmas: the comparison arguments for sign patterns other than the two extremal families could be collected into a single lemma rather than scattered across several short lemmas.
Simulated Author's Rebuttal
We thank the referee for the careful reading and positive assessment of the manuscript. The report correctly summarizes the main result: the proof of the Holevo–Utkin conjecture for all d ≥ 4, with the extremal ratio attained precisely on the two families of vectors identified in the abstract. We appreciate the recommendation for minor revision and will incorporate any editorial improvements in the revised version.
Circularity Check
No significant circularity
full rationale
The manuscript is a direct mathematical proof of an externally stated conjecture from Holevo and Utkin (cited as HU26, with d=3 case already proved there). The derivation proceeds by explicit case analysis: it reduces the ratio optimization (under zero-sum and unit-l2 constraints) to the two candidate vector families via direct comparison and strict convexity/concavity of |t|^p, without any self-definition of the target quantity, without fitting parameters to data and relabeling them as predictions, and without load-bearing self-citations or imported uniqueness theorems from the present authors' prior work. The argument for d≥4 is self-contained against the external conjecture and uses only standard analytic properties.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math ℓ_p quasi-norm satisfies the usual scaling and triangle inequality for p > 0
- standard math The minimum or maximum of a continuous function on a compact set is attained
Lean theorems connected to this paper
-
IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Lagrange multiplier equation p|xi|^{p-2} xi = λ + 2μ xi; reduction to vectors with at most two distinct values via strict convexity/concavity of t↦|t|^p
-
IndisputableMonolith/Foundation/AlphaCoordinateFixation.leanJ_uniquely_calibrated_via_higher_derivative unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Endpoint values Ψ(0)=2^{1-p/2} and Ψ(1)=((d-1)^{p/2}+(d-1)^{1-p/2})/d^{p/2} attained exactly by the two-support sign-alternating and equal-magnitude families
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
Works this paper leans on
-
[1]
P. Diaconis and L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab. 6, no. 3 (1996), 695--750
work page 1996
- [2]
- [3]
-
[4]
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
-
[5]
M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, 10th anniversary ed., Cambridge University Press, Cambridge, 2010
work page 2010
-
[6]
E. H. Lieb, Proof of an entropy conjecture of Wehrl, Comm. Math. Phys. 62, no. 1 (1978), 35--41
work page 1978
-
[7]
R. L. Frank, Sharp inequalities for coherent states and their optimizers, Adv. Nonlinear Stud. 23, no. 1 (2023), 20220050
work page 2023
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.