Suppose that $H$ is an Hilbert space with inner product $\inner{\cdot}{\cdot}$, and $L$ is a bounded adjoint operator on $H$. For any $\lambda\in \mathbb{R}$, we define the set $V_\lambda$ such that $$ V_\lambda = \{ x\in H\,:\, \inner{ Lx}{x } = \lambda \}.$$ Question: if $\lambda_1<\lambda_2<\lambda_3$, do we have $$ dist\{ V_{\lambda_2}, V_{\lambda_1} \} < dist\{ V_{\lambda_3}, V_{\lambda_1} \}? $$