Holevo–Utkin conjecture on sharp $\ell_p$ norms for zero-sum vectors

Let $d\ge 3$, and let $\|x\|_p$ denote the $\ell_p$ (quasi-)norm of a vector $x\in\mathbb R^d$. We consider nonzero zero-sum vectors, so that $\sum_{i=1}^d x_i=0$.

Holevo–Utkin conjecture. 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 $2<q<\infty$,

$$\max\left\{\frac{\|x\|_q}{\|x\|_2}:\vec{0}\neq x\in\mathbb R^d,\ \sum_{i=1}^d x_i=0\right\}=\max\left\{2^{1/q-1/2},\left(\frac{(d-1)^{q/2}+(d-1)^{1-q/2}}{d^{q/2}}\right)^{1/q}\right\}.$$

Holevo and Utkin proved the case $d=3$ in their earlier work. The present paper confirms the remaining cases $d\ge 4$, so the conjecture is solved. The two corresponding optimizer families are $(1,-1,0,\dots,0)$ and $(d-1,-1,-1,\dots,-1)$, up to permutation and a global sign.