A Sharp inequality between local volumes and minimal log discrepancies

We answer a question of Li–Liu–Xu: every $n$-dimensional klt germ $x\in (X,\mathrm{\Delta })$, where $n\ge 2$, satisfies the sharp inequality $$\stackrel{}{\mathrm{vol}}(x,X,\mathrm{\Delta })\le n^{n-1}{\mathrm{mld}}_x(X,\mathrm{\Delta }),$$ with equality if and only if $\mathrm{\Delta }=0$ near $x$, and analytically, $(x\in X)\cong \frac{1}{r}(1,\dots ,1)$ for some $r\ge 1$. We also prove that, in fixed dimension and with coefficients in a fixed finite set, $\stackrel{}{\mathrm{vol}}\mathrm{/}\mathrm{mld}$ is discrete away from zero. As applications of the sharp inequality, we obtain lower bounds for minimal log discrepancies of log Fano pairs.