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

TL;DR

This paper proves the remaining cases of the Holevo--Utkin conjecture on sharp $\, ext{ell}_p$ norms for zero-sum vectors in dimensions four and higher, completing the previously verified cases for three dimensions.

Contribution

The paper confirms the Holevo--Utkin conjecture for all dimensions $d \,\geq 4$, extending prior results that only covered the case $d=3$.

Findings

Confirmed the conjecture for $d \,\geq 4$ dimensions.

Provided explicit formulas for $\,\ell_p$ norm ratios for zero-sum vectors.

Extended the validity of the conjecture beyond the previously proven case $d=3$.

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 $d\ge 4$.