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

Mathematics > Classical Analysis and ODEs Title:Proof of the Holevo-Utkin conjecture on sharp $\ell_p$ norms for zero-sum vectors View PDF HTML (experimental)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 $d\ge 4$. Current browse context: Bibliographic and Citation Tools Code, Data and Media Associated with this Article Demos Recommenders and Search Tools arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.