Total log canonical thresholds and generalized Eckardt points

Let $X$ be a smooth hypersurface of degree $n\geqslant 3$ in ${\mathbb P}^n$. It is proved that the log canonical threshold of an arbitrary hyperplane section $H$ of it is at least $(n-1)/n$. Under the assumption of the log minimal model program it is also proved that the log canonical threshold of $H\subset X$ is $(n-1)/n$ if and only if $H$ is a cone in ${\mathbb P}^{n-1}$ over a smooth hypersurface of degree $n$ in ${\mathbb P}^{n-2}$.