Proof of the Holevo--Utkin conjecture on sharp ${\mathrm{\ell }}_p$ norms for zero-sum vectors

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