Let $A$ be a $n\times n$-matrix. We let $\|A\|_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is, $$ \|A\|_p = \sup_{x\neq 0}\frac{\|Ax\|_p}{\|x\|_p}. $$

By the Riesz-Thorin theorem, the $p$-norm of $A$, as a function of $p$, is log-convex, meaning that the function $[1,\infty]\ni p\mapsto \log(\|A\|_p)$ is convex.

Question:Is the p-norm of a matrixstrictlylog-convex (unless it is constant)?

The proof of the Riesz-Thorin theorem uses holomorphic functions, so it seems plausible that if $\log(\|A\|_p)$ is constant on some non-trivial interval, then $\log(\|A\|_p)$ has to be globally constant.

**Edit (following Terry Tao's answer):**

Can $\|A\|_p$ be constant on some non-trivial interval, yet not be globally constant?